Multilinear Algebra & Tensors | Advanced Topics | Linear Algebra

1. Multilinear Maps

Definition 8.6: Bilinear Map

A map $B: V \times W \to U$ is bilinear if:

$B(\alpha v_1 + \beta v_2, w) = \alpha B(v_1, w) + \beta B(v_2, w)$
$B(v, \alpha w_1 + \beta w_2) = \alpha B(v, w_1) + \beta B(v, w_2)$

Example 8.5: Examples of Bilinear Maps

Dot product: $\langle \cdot, \cdot \rangle: V \times V \to \mathbb{R}$
Matrix multiplication: $M_{m \times n} \times M_{n \times p} \to M_{m \times p}$
Cross product: $\mathbb{R}^3 \times \mathbb{R}^3 \to \mathbb{R}^3$

Definition 8.7: Multilinear Map

A k-linear map $T: V_1 \times \cdots \times V_k \to U$ is linear in each argument separately.

Example 8.5a: Determinant as Multilinear Map

The determinant det: (ℝⁿ)ⁿ → ℝ is n-linear and alternating:

\det(v_1, ..., \alpha v_i + \beta w, ..., v_n) = \alpha \det(v_1, ..., v_i, ..., v_n) + \beta \det(v_1, ..., w, ..., v_n)

Remark 8.5a: Bilinear ≠ Linear

A bilinear map is NOT linear as a map V × W → U. For example:

B(2v, w) = 2B(v, w) \neq B(v+v, w+w) = 2B(v,w) + 2B(v,w)

Bilinear means linear in each slot separately.

Theorem 8.5a: Bilinear Map Representation

If $\{e_i\}$ is basis for V and $\{f_j\}$ for W, then B(v, w) is determined by:

B(v, w) = \sum_{i,j} v^i w^j B(e_i, f_j)

A bilinear form on ℝⁿ is represented by an n×n matrix.

Example 8.5b: Bilinear Form Computation

For $B(v, w) = v^T A w$ with $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ :

B((1, 0)^T, (1, 1)^T) = (1, 0) \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \begin{pmatrix} 1 \\ 1 \end{pmatrix} = 3

2. Tensor Product

Definition 8.8: Tensor Product Space

The tensor product $V \otimes W$ is a vector space with a bilinear map $\otimes: V \times W \to V \otimes W$ such that any bilinear map $B: V \times W \to U$ factors uniquely through $V \otimes W$ .

Theorem 8.6: Dimension of Tensor Product

\dim(V \otimes W) = \dim(V) \cdot \dim(W)

If $\{e_i\}$ is basis for $V$ and $\{f_j\}$ for $W$ , then $\{e_i \otimes f_j\}$ is basis for $V \otimes W$ .

Example 8.6: Outer Product

For $u \in \mathbb{R}^m, v \in \mathbb{R}^n$ , the outer product $u \otimes v = uv^T$ is an $m \times n$ matrix:

(u \otimes v)_{ij} = u_i v_j

Example 8.6a: Concrete Outer Product

For $u = (1, 2)^T$ and $v = (3, 4, 5)^T$ :

u \otimes v = \begin{pmatrix} 1 \\ 2 \end{pmatrix} \begin{pmatrix} 3 & 4 & 5 \end{pmatrix} = \begin{pmatrix} 3 & 4 & 5 \\ 6 & 8 & 10 \end{pmatrix}

Remark 8.6a: Universal Property

The tensor product is characterized by its universal property:

For any bilinear B: V × W → U, there exists a unique linear L: V ⊗ W → U such that B(v, w) = L(v ⊗ w).

Theorem 8.6a: Properties of Tensor Product

$(v_1 + v_2) \otimes w = v_1 \otimes w + v_2 \otimes w$
$v \otimes (w_1 + w_2) = v \otimes w_1 + v \otimes w_2$
$(\alpha v) \otimes w = v \otimes (\alpha w) = \alpha (v \otimes w)$

Definition 8.8a: Simple vs General Tensors

Simple tensors (or decomposable) have form v ⊗ w.

General tensors are sums of simple tensors: $\sum_i v_i \otimes w_i$ .

Not every tensor is simple! Most tensors require multiple terms.

Example 8.6b: Non-Simple Tensor

In $\mathbb{R}^2 \otimes \mathbb{R}^2$ , the tensor:

e_1 \otimes e_1 + e_2 \otimes e_2

cannot be written as a single u ⊗ v (verify by trying to factor).

Remark 8.6b: SVD Connection

The SVD A = UΣVᵀ expresses a matrix as sum of simple tensors:

A = \sum_{i=1}^r \sigma_i u_i \otimes v_i

Rank = minimum number of simple tensors needed.

3. Einstein Summation Convention

Definition 8.9: Einstein Convention

Repeated indices (one upper, one lower) imply summation:

a_i b^i \equiv \sum_i a_i b^i, \quad T^i{}_j v^j \equiv \sum_j T^i{}_j v^j

Remark 8.6: Index Position

Upper indices ( $v^i$ ): contravariant (vector components)
Lower indices ( $v_i$ ): covariant (dual/covector components)

Example 8.7: Matrix-Vector Product

Using Einstein notation: $(Av)^i = A^i{}_j v^j$

The $j$ index is summed (appears up and down).

Example 8.7a: Matrix Multiplication

Product C = AB in Einstein notation:

C^i{}_k = A^i{}_j B^j{}_k

Index j appears up and down, so we sum over j.

Example 8.7b: Trace

The trace of a matrix:

\text{tr}(A) = A^i{}_i = \sum_i A^i{}_i

Definition 8.9a: Index Raising and Lowering

The metric tensor $g_{ij}$ converts between index types:

Lower: $v_i = g_{ij} v^j$
Raise: $v^i = g^{ij} v_j$

where $g^{ij}$ is inverse of $g_{ij}$ .

Example 8.7c: Euclidean Metric

In Euclidean space with orthonormal basis: $g_{ij} = \delta_{ij}$ (Kronecker delta).

So $v_i = v^i$ — no distinction between upper and lower indices.

Remark 8.7: Free vs Dummy Indices

Free indices: Appear once, indicate tensor type
Dummy indices: Repeated (summed), can be renamed

In $T^i{}_j v^j$ : i is free, j is dummy.

4. Tensors

Definition 8.10: (p,q)-Tensor

A (p,q)-tensor is a multilinear map:

T: \underbrace{V^* \times \cdots \times V^*}_{p} \times \underbrace{V \times \cdots \times V}_{q} \to \mathbb{R}

It has $p$ contravariant and $q$ covariant indices: $T^{i_1 \ldots i_p}{}_{j_1 \ldots j_q}$ .

Example 8.8: Common Tensors

(0,0): Scalar
(1,0): Vector $v^i$
(0,1): Covector $\omega_i$
(1,1): Linear map $A^i{}_j$
(0,2): Bilinear form $g_{ij}$ (metric)

Definition 8.10a: Tensor Transformation Law

Under change of basis $e'_i = P^j{}_i e_j$ , a (p,q)-tensor transforms as:

T'^{i_1 \ldots i_p}{}_{j_1 \ldots j_q} = (P^{-1})^{i_1}{}_{k_1} \cdots P^{l_1}{}_{j_1} \cdots T^{k_1 \ldots}{}_{l_1 \ldots}

This transformation law DEFINES what a tensor is!

Definition 8.10b: Tensor Contraction

Contraction sets one upper and one lower index equal and sums:

T^{ij}{}_{jk} \to \sum_j T^{ij}{}_{jk} = S^i{}_k

Reduces rank by 2 (one upper, one lower index removed).

Example 8.8a: Contraction Examples

Trace: $A^i{}_i$ contracts a (1,1)-tensor to scalar
Dot product: $v^i w_i$ contracts two vectors (one raised)
Matrix-vector: $A^i{}_j v^j$ contracts (1,1)⊗(1,0) to (1,0)

Definition 8.10c: Tensor Product of Tensors

If S is (p₁,q₁) and T is (p₂,q₂), then S⊗T is (p₁+p₂, q₁+q₂):

(S \otimes T)^{i_1 \ldots i_{p_1} j_1 \ldots j_{p_2}}{}_{k_1 \ldots l_1 \ldots} = S^{i_1 \ldots}{}_{k_1 \ldots} T^{j_1 \ldots}{}_{l_1 \ldots}

Remark 8.8: Tensor Algebra

Tensors form an algebra with:

Addition: Same type tensors
Scalar multiplication: Multiply all components
Tensor product: Combines tensors
Contraction: Reduces rank

Definition 8.10d: Symmetric and Antisymmetric Tensors

A (0,2)-tensor is:

Symmetric: $T_{ij} = T_{ji}$
Antisymmetric: $T_{ij} = -T_{ji}$

Any tensor can be decomposed: $T = T^{sym} + T^{anti}$

Example 8.8b: Symmetrization

For any (0,2)-tensor T:

T^{sym}_{ij} = \frac{1}{2}(T_{ij} + T_{ji}), \quad T^{anti}_{ij} = \frac{1}{2}(T_{ij} - T_{ji})

5. Applications

General Relativity

Metric tensor, Riemann curvature, Einstein field equations—all tensor equations.

Continuum Mechanics

Stress tensor, strain tensor, elasticity tensor relate forces and deformations.

Machine Learning

Data as tensors (images, sequences). Tensor decomposition for compression and analysis.

Quantum Mechanics

State spaces as tensor products. Entanglement is non-separability in tensor product.

Example 8.9: Stress Tensor

The stress tensor $\sigma_{ij}$ relates force to surface orientation:

F_i = \sigma_{ij} n^j \cdot A

where $n^j$ is surface normal and A is area. It's a (0,2) symmetric tensor.

Example 8.9a: Moment of Inertia

The inertia tensor $I_{ij}$ relates angular momentum to angular velocity:

L^i = I^{ij} \omega_j

Diagonalizing I gives principal axes and principal moments.

Example 8.9b: Metric Tensor

The metric tensor $g_{ij}$ defines inner product and distance:

ds^2 = g_{ij} dx^i dx^j, \quad \langle u, v \rangle = g_{ij} u^i v^j

In general relativity, the metric encodes gravitational effects.

Remark 8.9: Physics Notation

Physicists often write tensors with explicit indices:

$g_{\mu\nu}$ — spacetime metric (Greek indices: 0-3)
$T_{\mu\nu}$ — stress-energy tensor
$F_{\mu\nu}$ — electromagnetic field tensor

Example 8.9c: Machine Learning Tensors

In deep learning frameworks:

1D tensor: Vector (features)
2D tensor: Matrix (batch of vectors)
3D tensor: Grayscale video or batch of images
4D tensor: Batch of color images (N×H×W×C)

6. Levi-Civita Symbol

Definition 8.11: Levi-Civita Symbol

The Levi-Civita symbol $\varepsilon_{ijk}$ (in 3D):

\varepsilon_{ijk} = \begin{cases} +1 & \text{if } (i,j,k) \text{ is even permutation of } (1,2,3) \\ -1 & \text{if } (i,j,k) \text{ is odd permutation} \\ 0 & \text{if any index repeated} \end{cases}

Example 8.10: Cross Product

The cross product in index notation:

(u \times v)^i = \varepsilon^{ijk} u_j v_k

Example 8.10a: Determinant

The 3×3 determinant:

\det(A) = \varepsilon^{ijk} A_{1i} A_{2j} A_{3k}

Remark 8.10: Pseudo-tensors

The Levi-Civita symbol is a pseudo-tensor:

Transforms like a tensor under rotations
Gets extra (-1) under reflections
Cross product is a pseudo-vector (axial vector)

7. Key Concepts Summary

Tensor Product

• $\dim(V \otimes W) = mn$
• $v \otimes w$ : simple tensor
• Outer product: $uv^T$

Index Notation

• Upper: contravariant
• Lower: covariant
• Repeated: sum (Einstein)

Tensor Types Table

Type	Components	Example
(0,0)	1	Scalar
(1,0)	n	Vector vⁱ
(0,1)	n	Covector ωᵢ
(1,1)	n²	Linear map Aⁱⱼ
(0,2)	n²	Metric gᵢⱼ
(p,q)	n^(p+q)	General tensor

8. Tensor Decompositions

Definition 8.12: Tensor Rank

The rank of a tensor T is the minimum number of simple tensors needed:

T = \sum_{r=1}^R v_r^{(1)} \otimes v_r^{(2)} \otimes \cdots

For matrices, this is ordinary matrix rank. For higher-order tensors, it's more complex.

Remark 8.11: Tensor Decomposition Types

CP/CANDECOMP: Sum of rank-1 tensors
Tucker: Core tensor with factor matrices
Tensor Train: Chain of 3D tensors

Example 8.11: Matrix as 2-Tensor

SVD is a tensor decomposition for 2-tensors (matrices):

A = \sum_{i=1}^r \sigma_i u_i \otimes v_i

This is the CP decomposition for matrices.

Remark 8.12: Applications of Tensor Decomposition

Data compression: Approximate high-dimensional data
Recommendation systems: Collaborative filtering
Scientific computing: PDE solvers
Quantum chemistry: Electronic structure

9. Common Mistakes

Confusing tensor and matrix

A matrix is a specific representation. A tensor is defined by transformation laws. Different bases give different matrices for same tensor.

Wrong index contraction

Can only contract one upper with one lower index. Tⁱʲ cannot contract with Tᵏˡ unless indices match position.

Thinking all tensors are simple

Most tensors require sums of simple tensors. e₁⊗e₁ + e₂⊗e₂ cannot be written as u⊗v.

10. Chapter Summary

Key Takeaways

Core Concepts

• Tensors: multilinear maps
• V⊗W: tensor product space
• Einstein notation: implied sums
• (p,q)-tensor: p upper, q lower indices

Operations

• Tensor product: combines tensors
• Contraction: reduces rank by 2
• Index raising/lowering: via metric
• Symmetrization/antisymmetrization

Remark 8.13: Looking Forward

Tensors lead to:

Differential geometry: Riemannian manifolds, curvature
Physics: General relativity, continuum mechanics
Machine learning: Tensor networks, attention mechanisms
Category theory: Monoidal categories

Pro Tips

• Einstein convention: repeated index (one up, one down) means sum
• dim(V⊗W) = dim(V) × dim(W)
• Contraction reduces rank by 2
• Tensor = defined by transformation law
• Use metric to raise/lower indices

11. More Worked Examples

Example 8.12: Tensor Product Basis

Let V = ℝ² with basis {e₁, e₂} and W = ℝ³ with basis {f₁, f₂, f₃}.

Basis for V⊗W (dimension 6):

\{e_1 \otimes f_1, e_1 \otimes f_2, e_1 \otimes f_3, e_2 \otimes f_1, e_2 \otimes f_2, e_2 \otimes f_3\}

Example 8.12a: Expanding a Tensor

Express $(2e_1 + 3e_2) \otimes (f_1 - f_2)$ in the basis:

= 2(e_1 \otimes f_1) - 2(e_1 \otimes f_2) + 3(e_2 \otimes f_1) - 3(e_2 \otimes f_2)

Example 8.12b: Contraction Calculation

For $T^{ij}{}_{k}$ with components, compute the contraction $T^{i1}{}_{1}$ :

S^i = T^{ij}{}_{j} = \sum_j T^{ij}{}_{j} = T^{i1}{}_{1} + T^{i2}{}_{2} + \cdots

Example 8.12c: Metric Tensor Example

In 2D with polar coordinates, the metric tensor is:

g_{ij} = \begin{pmatrix} 1 & 0 \\ 0 & r^2 \end{pmatrix}

Line element: $ds^2 = dr^2 + r^2 d\theta^2$

Example 8.12d: Electromagnetic Field Tensor

The electromagnetic field tensor $F_{\mu\nu}$ is antisymmetric (0,2)-tensor:

F_{\mu\nu} = \begin{pmatrix} 0 & E_x & E_y & E_z \\ -E_x & 0 & -B_z & B_y \\ -E_y & B_z & 0 & -B_x \\ -E_z & -B_y & B_x & 0 \end{pmatrix}

Maxwell's equations become tensor equations!

12. Exterior Algebra (Preview)

Definition 8.13: Wedge Product

The wedge product (exterior product) is antisymmetric:

v \wedge w = -w \wedge v, \quad v \wedge v = 0

Remark 8.14: Exterior vs Tensor Product

Tensor: v⊗w ≠ w⊗v in general
Exterior: v∧w = -w∧v (antisymmetric)
Symmetric: v⊙w = w⊙v (symmetric product)

Example 8.13: Wedge Product in ℝ³

For $u = e_1 + e_2$ and $v = e_2 + e_3$ :

u \wedge v = (e_1 + e_2) \wedge (e_2 + e_3) = e_1 \wedge e_2 + e_1 \wedge e_3 + e_2 \wedge e_3

Note: e₂∧e₂ = 0 vanishes.

Remark 8.15: Differential Forms

Exterior algebra is the foundation of differential forms:

0-forms: functions
1-forms: dx, dy, dz
2-forms: dx∧dy, etc.
3-forms: dx∧dy∧dz (volume)

13. Quick Reference

Index Notation Rules

• Free indices: appear once, same on both sides
• Dummy indices: appear twice, summed
• Upper + lower: can contract
• Metric: raises/lowers indices

Common Operations

• Dot: vⁱwᵢ = scalar
• Outer: vⁱwʲ = (1,1)-tensor
• Trace: Aⁱᵢ = scalar
• Matrix: Aⁱⱼvʲ = vector

Notation Summary

V ⊗ W	Tensor product of spaces
v ⊗ w	Simple tensor (outer product)
Tⁱⱼ	(1,1)-tensor components
gᵢⱼ	Metric tensor
εᵢⱼₖ	Levi-Civita symbol
δⁱⱼ	Kronecker delta

14. Practice Problems

Example 8.14: Practice Problem 1

Compute the dimension of V⊗W⊗U where dim(V)=2, dim(W)=3, dim(U)=4.

Solution: dim(V⊗W⊗U) = 2×3×4 = 24

Example 8.14a: Practice Problem 2

Write the trace of A in Einstein notation.

Solution: tr(A) = Aⁱᵢ = A¹₁ + A²₂ + ... + Aⁿₙ

Example 8.14b: Practice Problem 3

If Tⁱⱼ is a (1,1)-tensor and vʲ is a (1,0)-tensor, what is Tⁱⱼvʲ?

Solution: A (1,0)-tensor (vector). The j index is contracted.

Example 8.14c: Practice Problem 4

Show that εⁱʲᵏεᵢⱼₖ = 6 in ℝ³.

Solution: Sum over all permutations: each nonzero term contributes ±1×(±1) = 1, and there are 6 such terms (3! = 6).

Remark 8.16: Mastery Checklist

You've mastered tensors when you can:

✓ Define bilinear and multilinear maps
✓ Compute tensor products and contractions
✓ Use Einstein summation convention fluently
✓ Identify tensor types (p,q) from indices
✓ Apply tensors to physics examples

15. Tensors in Quantum Mechanics

Remark 8.17: Quantum State Spaces

In quantum mechanics, composite systems use tensor products:

Single particle: state space $\mathcal{H}$
Two particles: $\mathcal{H}_1 \otimes \mathcal{H}_2$
n particles: $\mathcal{H}^{\otimes n}$

Definition 8.14: Product States

A product state (separable) has form |ψ⟩ ⊗ |φ⟩.

An entangled state cannot be written as a single product.

Example 8.15: Bell State

The Bell state (maximally entangled):

|\Phi^+\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)

Cannot be written as |a⟩ ⊗ |b⟩ — it's entangled!

Remark 8.18: Quantum Computing

Quantum computers exploit tensor product structure:

n qubits: state in $(\mathbb{C}^2)^{\otimes n}$ , dim = 2ⁿ
Quantum gates: unitary operators on tensor space
Entanglement enables quantum speedup

16. Tensor Networks (Preview)

Definition 8.15: Tensor Network

A tensor network is a collection of tensors with some indices contracted between them, represented as a graph.

Remark 8.19: Applications

Quantum many-body: MPS, PEPS, MERA
Machine learning: Tensor train decomposition
Optimization: Efficient representation of high-dimensional data

Example 8.16: Matrix Product State

An MPS represents a many-body state efficiently:

|\psi\rangle = \sum_{i_1,...,i_n} A^{[1]}_{i_1} A^{[2]}_{i_2} \cdots A^{[n]}_{i_n} |i_1...i_n\rangle

Each A^[k] is a matrix, and we multiply them to get coefficients.

17. Historical Context

Remark 8.20: History

Key developments in tensor theory:

1890s: Ricci-Curbastro develops tensor calculus
1915: Einstein uses tensors for general relativity
1920s: Tensor notation becomes standard in physics
Modern: Tensor networks in quantum computing and ML

The Power of Tensors

Tensors provide a coordinate-independent way to describe physical laws. Einstein famously said the happiest thought of his life was realizing that physical laws should take the same form in all coordinate systems — which requires tensor equations.

18. Final Synthesis

The Big Picture

Tensors generalize vectors and matrices to arbitrarily many indices. They provide the natural language for:

Physics

Relativity, electromagnetism, continuum mechanics — all tensor theories.

Mathematics

Differential geometry, algebraic topology, representation theory.

Computing

Deep learning, quantum computing, data science — all tensor computations.

Remark 8.21: Where Next?

From here, explore:

Applications (LA-8.4): See tensors in action
Differential geometry: Manifolds and curvature
Representation theory: Tensors and group actions
Numerical multilinear algebra: Computational methods

19. Additional Theory

Definition 8.16: Kronecker Product

For matrices A (m×n) and B (p×q), the Kronecker product A⊗B is (mp×nq):

A \otimes B = \begin{pmatrix} a_{11}B & \cdots & a_{1n}B \\ \vdots & \ddots & \vdots \\ a_{m1}B & \cdots & a_{mn}B \end{pmatrix}

Example 8.17: Kronecker Product Example

For $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ and $B = \begin{pmatrix} 0 & 5 \\ 6 & 7 \end{pmatrix}$ :

A \otimes B = \begin{pmatrix} 0 & 5 & 0 & 10 \\ 6 & 7 & 12 & 14 \\ 0 & 15 & 0 & 20 \\ 18 & 21 & 24 & 28 \end{pmatrix}

Theorem 8.7: Kronecker Product Properties

$(A \otimes B)(C \otimes D) = (AC) \otimes (BD)$ (if sizes match)
$(A \otimes B)^{-1} = A^{-1} \otimes B^{-1}$
$\text{tr}(A \otimes B) = \text{tr}(A) \cdot \text{tr}(B)$
$\det(A \otimes B) = (\det A)^p (\det B)^m$

Remark 8.22: Vectorization

The vec operation stacks columns of a matrix into a vector:

\text{vec}(ABC) = (C^T \otimes A) \text{vec}(B)

This connects matrix equations to Kronecker products.

Definition 8.17: Hadamard Product

The Hadamard product (element-wise) for same-sized matrices:

(A \circ B)_{ij} = a_{ij} b_{ij}

20. Computational Aspects

Tensor Libraries

Library	Language	Focus
NumPy	Python	General arrays, einsum
PyTorch	Python	Deep learning, autograd
TensorFlow	Python	Deep learning, TPU support
ITensor	C++/Julia	Tensor networks, physics

Example 8.18: NumPy einsum

Einstein summation in NumPy:

Matrix multiply: np.einsum('ij,jk->ik', A, B)
Trace: np.einsum('ii->', A)
Outer: np.einsum('i,j->ij', u, v)

Remark 8.23: Computational Complexity

Tensor operations can be expensive:

Storage: n^k for order-k tensor
Contraction: O(n^(k+l-2)) for two tensors
Decomposition: often O(n^k) or higher

Tensor networks provide efficient representations!

21. Additional Practice

Example 8.19: Practice Problem 5

Compute $(A \otimes I)(I \otimes B)$ for 2×2 matrices A, B.

Solution: By Kronecker product property:

(A \otimes I)(I \otimes B) = (AI) \otimes (IB) = A \otimes B

Example 8.19a: Practice Problem 6

If T is a (2,1)-tensor in ℝ³, how many components does it have?

Solution: 3² × 3 = 27 components

Example 8.19b: Practice Problem 7

Show that the trace is basis-independent.

Solution: Under change of basis P:

\text{tr}(P^{-1}AP) = \text{tr}(AP \cdot P^{-1}) = \text{tr}(A)

Tensors Practice

12

Questions

0

Correct

0%

Accuracy

1

A bilinear map

B: V \times W \to U

is linear in:

Easy

Not attempted

2

The tensor product

v \otimes w

is:

Easy

Not attempted

3

If

\dim V = m

and

\dim W = n

, then

\dim(V \otimes W) =

Medium

Not attempted

4

The outer product of column vectors

u

and

v

is:

Easy

Not attempted

5

A

(2,1)

-tensor has:

Medium

Not attempted

6

Einstein summation convention means:

Medium

Not attempted

7

The stress tensor in physics is:

Medium

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8

Tensor contraction reduces rank by:

Medium

Not attempted

9

Which is NOT a tensor operation?

Hard

Not attempted

10

In machine learning, a 3D tensor might represent:

Easy

Not attempted

11

The metric tensor

g_{ij}

is used to:

Hard

Not attempted

12

A tensor is characterized by its:

Hard

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Frequently Asked Questions

What is a tensor?

A tensor is a multilinear map from products of vector spaces (and their duals) to scalars. Equivalently, it's an array of numbers that transforms in a specific way under change of basis.

How does a tensor differ from a matrix?

A matrix is a 2D array, specifically a (1,1)-tensor or a representation of a linear map. Tensors generalize to any number of indices and include specific transformation rules.

What is the tensor product?

V ⊗ W is a new vector space such that bilinear maps from V × W correspond to linear maps from V ⊗ W. Elements v ⊗ w are 'simple tensors'; general elements are sums of these.

Why is Einstein notation useful?

It compactly expresses sums over indices. Instead of Σᵢ aᵢbⁱ, write aᵢbⁱ. Repeated indices (one up, one down) imply summation. Makes tensor equations cleaner.

What are covariant vs contravariant?

Contravariant (upper indices) transform opposite to basis vectors. Covariant (lower indices) transform like basis vectors. The distinction matters for non-orthonormal bases.

How are tensors used in physics?

Stress, strain, inertia, electromagnetic field, metric in general relativity—all tensors. They express physical laws independent of coordinate choice.

How are tensors used in machine learning?

Data is stored in tensors: images (H×W×C), batches (N×H×W×C), sequences (N×T×D). Deep learning frameworks (PyTorch, TensorFlow) are built around tensor operations.

What is tensor contraction?

Setting one upper and one lower index equal and summing: Tⁱⱼₖ → Tⁱⱼᵢ = Σᵢ Tⁱⱼᵢ. Reduces rank by 2. Matrix trace is a contraction.

What's the relationship to the determinant?

The determinant can be expressed using the Levi-Civita tensor εⁱʲᵏ. det(A) = εⁱʲᵏ a₁ᵢ a₂ⱼ a₃ₖ. This connects determinants to multilinear algebra.

Is the cross product a tensor?

The cross product is actually a pseudovector (axial vector). It uses the Levi-Civita symbol and behaves differently under reflections than true vectors.