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Appendix B Problems

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Table of Contents

Basic

Medium

Advanced

Basic

B-1. Basics of the Tensor Product \(S \otimes T\)

Let \(V\) be a 2-dimensional linear space with basis \(e_1, e_2\). Given \(S = 3e_1 - e_2\) and \(T = e_1 + 4e_2\), expand the tensor product \(S \otimes T\) in terms of the basis \(\{e_i \otimes e_j\}\) of \(T^2(V)\).

Hint

Use the distributive law \((a e_1 + b e_2) \otimes (c e_1 + d e_2) = ac\,e_1 \otimes e_1 + ad\,e_1 \otimes e_2 + bc\,e_2 \otimes e_1 + bd\,e_2 \otimes e_2\).

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B-2. Non-commutativity of the Tensor Product

Let \(V\) be a 2-dimensional linear space with basis \(e_1, e_2\). Given \(S = 3e_1 - e_2\) and \(T = e_1 + 4e_2\), expand \(T \otimes S\) in the basis of \(T^2(V)\), and verify that \(S \otimes T\) and \(T \otimes S\) are different.

Hint

The tensor product is generally non-commutative: \(S \otimes T \neq T \otimes S\). Compare the coefficients of \(e_1 \otimes e_2\) and \(e_2 \otimes e_1\).

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B-3. Linear Combinations of Tensors

For elements \(P = 2\,e_1 \otimes e_1 - 3\,e_1 \otimes e_2 + e_2 \otimes e_1 + 5\,e_2 \otimes e_2\) and \(Q = -e_1 \otimes e_1 + 4\,e_1 \otimes e_2 - e_2 \otimes e_1 + 2\,e_2 \otimes e_2\) in \(T^2(V)\) (where \(V\) is 2-dimensional), find \(P + Q\) and \(2P - Q\) respectively.

Hint

Add or subtract the coefficients of the same basis elements \(e_i \otimes e_j\). This is the same operation as linear combinations of ordinary vectors.

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B-4. Dimensions of Tensor Spaces

When \(V\) is 4-dimensional, find the dimensions of \(T^2(V)\) and \(T^3(V)\) respectively.

Hint

When \(V\) is \(n\)-dimensional, the dimension of \(T^r(V)\) is \(n^r\).

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B-5. Expansion of Einstein Summation

In the expression \(S = S^{ij}\,e_i \otimes e_j\) written using Einstein's summation convention, when \(V\) is 3-dimensional (\(i, j = 1, 2, 3\)), write this expression in the explicit form of a double sum using the \(\Sigma\) notation. Furthermore, state how many terms appear when fully expanded.

Hint

Write \(S = \sum_{i=1}^{3}\sum_{j=1}^{3} S^{ij}\,e_i \otimes e_j\) and count all combinations of \(i\) and \(j\).

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B-6. Violations of the Einstein Convention

Select those that violate the conventions of Einstein summation notation.

  • (a) \(A^i B^j\,e_i \otimes e_j\)
  • (b) \(A^i B^i C^i\)
  • (c) \(S^{ij}\,T_{jk}\)
  • (d) \(A^i B_j\)
  • (e) \(S^{ij}\,e_i \otimes e_i\)
Hint

Rules of the convention: (1) When the same index appears once as an upper index and once as a lower index, summation is implied; (2) The same index must not appear three or more times in a single term; (3) Free indices (indices not summed over) must match across all terms.

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B-7. Reconstructing a Tensor from Components

Given that the components of an element \(S\) of \(T^2(V)\) (where \(V\) is 2-dimensional) are \(S^{11} = 2,\; S^{12} = -1,\; S^{21} = 0,\; S^{22} = 3\), write \(S\) explicitly using the basis \(e_i \otimes e_j\).

Hint

Expand the contraction \(S = S^{ij}\,e_i \otimes e_j\) and substitute the values of the components.

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B-8. Tensor Product of Rank 1 and Rank 2 Tensors

Let \(V\) be a 2-dimensional linear space. Expand the tensor product \(A \otimes S\) of \(A = 2e_1 + e_2 \in T^1(V)\) and \(S = e_1 \otimes e_2 - 3\,e_2 \otimes e_1 \in T^2(V)\) in terms of the basis of \(T^3(V)\).

Hint

Since \(A \otimes S\) is the tensor product of an element of \(T^1(V)\) and an element of \(T^2(V)\), it is an element of \(T^{1+2}(V) = T^3(V)\). Expand using the distributive law.

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Medium

M-1. Condition for Decomposable Tensors

Let \(V\) be a 2-dimensional linear space with basis \(e_1, e_2\). Show that an element \(W = a\,e_1 \otimes e_1 + b\,e_1 \otimes e_2 + c\,e_2 \otimes e_1 + d\,e_2 \otimes e_2\) of \(T^2(V)\) is a decomposable tensor (i.e., \(W = S \otimes T\) for some \(S, T \in V\)) if and only if \(ad - bc = 0\).

Hint

Set \(S = \alpha e_1 + \beta e_2\) and \(T = \gamma e_1 + \delta e_2\), expand \(S \otimes T\), and compare \(ad\) with \(bc\) using \(a = \alpha\gamma,\; b = \alpha\delta,\; c = \beta\gamma,\; d = \beta\delta\). Also show the converse direction (if \(ad = bc\) then the tensor is decomposable).

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M-2. Component Representation of Bilinear Maps

Let \(V\) be a 2-dimensional linear space with basis \(e_1, e_2\). Given that the components of a \(\binom{0}{2}\) tensor \(f\) are \(f_{11}=2, f_{12}=1, f_{21}=1, f_{22}=3\), answer the following.

(a) For general vectors \(\vec{A} = A^1 e_1 + A^2 e_2\) and \(\vec{B} = B^1 e_1 + B^2 e_2\), express \(f(\vec{A}, \vec{B})\) in terms of the components \(A^i, B^j, f_{ij}\) using multilinearity.

(b) When \(\vec{A} = e_1 - 2e_2\) and \(\vec{B} = 3e_1 + e_2\), find the value of \(f(\vec{A}, \vec{B})\).

(c) Verify using components that \(f(\vec{A}, \vec{B}) = f(\vec{B}, \vec{A})\) holds for arbitrary \(\vec{A}, \vec{B}\) for this \(f\).

Hint

In (a), derive \(f(\vec{A}, \vec{B}) = A^i B^j f_{ij}\) similarly to equation (B.7). In (c), confirm that \(f_{ij} = f_{ji}\).

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M-3. Rank Addition via Tensor Product

Show using component notation that the tensor product of an element of \(T^r(V)\) and an element of \(T^m(V)\) is an element of \(T^{r+m}(V)\). That is, for \(S = S^{i_1 \cdots i_r}\,e_{i_1} \otimes \cdots \otimes e_{i_r} \in T^r(V)\) and \(T = T^{j_1 \cdots j_m}\,e_{j_1} \otimes \cdots \otimes e_{j_m} \in T^m(V)\), show that \(S \otimes T\) can be expressed with components carrying \(r + m\) upper indices.

Hint

Use the distributive law of the tensor product and the scalar extraction properties (B.1)–(B.3) to expand \(S \otimes T\), and confirm that the basis takes the form \(e_{i_1} \otimes \cdots \otimes e_{i_r} \otimes e_{j_1} \otimes \cdots \otimes e_{j_m}\).

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M-4. \(T^0(V)\) and Scalars

Let \(V\) be an \(n\)-dimensional linear space. When \(T^0(V)\) is defined as the "rank-0 contravariant tensor space," its dimension is \(n^0 = 1\). Explain that elements of \(T^0(V)\) correspond to scalars (real numbers), and show that the tensor product \(\lambda \otimes S\) of any element \(S\) of \(T^r(V)\) with a scalar \(\lambda \in T^0(V)\) coincides with an element of \(T^r(V)\) (the scalar multiple \(\lambda S\)), using the tensor product rule (B.1).

Hint

\(T^0(V)\) is a 1-dimensional space whose only basis element is the "empty tensor product," and it can be identified with the real number field \(\mathbb{R}\). Apply (B.1) to \(\lambda \otimes S\).

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M-5. Simple \(S \otimes T\) Expansion

Let \(V\) be a 2-dimensional linear space with basis \(e_1, e_2\). Given \(S = e_1 + 2e_2\) and \(T = 3e_1 - e_2\), expand \(S \otimes T\) in terms of the basis of \(T^2(V)\).

Hint

Use the distributive law to expand \((e_1 + 2e_2) \otimes (3e_1 - e_2)\).

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M-6. Dimension of Tensor Spaces in 3-Dimensional Space

What is the dimension of \(T^2(V)\) (where \(V\) is 3-dimensional)? Also, what is the dimension of \(T^3(V)\)?

Hint

When \(V\) is \(n\)-dimensional, the dimension of \(T^r(V)\) is \(n^r\). Substitute \(n = 3\).

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M-7. Non-decomposability of Antisymmetric Tensors

Show that \(e_1 \otimes e_2 - e_2 \otimes e_1\) cannot be written as a decomposable tensor (i.e., in the form \(S \otimes T\)).

Hint

Setting \(a = 0, b = 1, c = -1, d = 0\), we get \(ad - bc = 0 - (-1) = 1 \neq 0\). Use the result of Problem B.9.

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M-8. Evaluation of the Identity Bilinear Form

Given that a \(\binom{0}{2}\) tensor \(f\) satisfies \(f(e_1, e_1) = 1\), \(f(e_1, e_2) = 0\), \(f(e_2, e_1) = 0\), \(f(e_2, e_2) = 1\), find \(f(\vec{A}, \vec{B})\) for \(\vec{A} = 3e_1 + 2e_2\) and \(\vec{B} = -e_1 + 4e_2\).

Hint

Substitute the components into \(f(\vec{A}, \vec{B}) = A^i B^j f_{ij}\). Note that \(f_{ij} = \delta_{ij}\) (the identity matrix).

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Advanced

A-1. Symmetric and Antisymmetric Decomposition

Define

\(W^{(S)} := \frac{1}{2}(W^{ij} + W^{ji})\,e_i \otimes e_j, \qquad W^{(A)} := \frac{1}{2}(W^{ij} - W^{ji})\,e_i \otimes e_j\)

(a) Show that any \(W \in T^2(V)\) can be uniquely decomposed as \(W = W^{(S)} + W^{(A)}\).

(b) Show that the set of all symmetric tensors \(\mathrm{Sym}^2(V) = \{W \in T^2(V) \mid W^{ij} = W^{ji}\}\) and the set of all antisymmetric tensors \(\mathrm{Alt}^2(V) = \{W \in T^2(V) \mid W^{ij} = -W^{ji}\}\) are each subspaces of \(T^2(V)\), and find the dimensions of \(\mathrm{Sym}^2(V)\) and \(\mathrm{Alt}^2(V)\) when \(V\) is \(n\)-dimensional.

(c) Given that the metric tensor \(g_{\mu\nu}\) is a symmetric tensor (\(g_{\mu\nu} = g_{\nu\mu}\)) and the electromagnetic field tensor (Faraday tensor) \(F_{\mu\nu}\) is an antisymmetric tensor (\(F_{\mu\nu} = -F_{\nu\mu}\)), find the number of independent components of \(g_{\mu\nu}\) and \(F_{\mu\nu}\) respectively in 4-dimensional spacetime (\(n = 4\)). Verify that their sum equals \(n^2\).

Hint

(b) The dimension of \(\mathrm{Sym}^2(V)\) is \(\frac{n(n+1)}{2}\), and the dimension of \(\mathrm{Alt}^2(V)\) is \(\frac{n(n-1)}{2}\). (c) Substitute \(n = 4\).

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A-2. Tensor Representation via Dual Spaces

(a) Write down the basis and dimension of the tensor product space \(V^* \otimes V^*\) of \(V^*\) and \(V^*\).

(b) Show that a \(\binom{0}{2}\) tensor \(f\) (a bilinear map that takes 2 vectors and returns a real number) can be expressed as an element of \(V^* \otimes V^*\) using the components \(f_{ij} = f(e_i, e_j)\):

\(f = f_{ij}\,e^i \otimes e^j\)

Demonstrate this by verifying that \(f(\vec{A}, \vec{B}) = A^k B^l f_{kl}\) is reproduced for arbitrary vectors \(\vec{A} = A^k e_k\), \(\vec{B} = B^l e_l\). Here, \((e^i \otimes e^j)(\vec{A}, \vec{B}) := e^i(\vec{A})\,e^j(\vec{B})\) is taken as the definition.

(c) Based on this result, discuss the fact that the space of \(\binom{0}{2}\) tensors is isomorphic to \(V^* \otimes V^*\), and that the space of \(\binom{0}{N}\) tensors is isomorphic to \(\underbrace{V^* \otimes \cdots \otimes V^*}_{N}\). Explain how the "connection between the two approaches" in Section B.11 of the main text generalizes.

Hint

(b) Substitute \((\vec{A}, \vec{B})\) into \(f_{ij}\,e^i \otimes e^j\) and use \(e^i(\vec{A}) = A^i\) (the definition of the dual basis). (c) In contrast to the contravariant tensor space \(T^r(V) = V \otimes \cdots \otimes V\), construct the argument for the covariant tensor space \(T_N(V) = V^* \otimes \cdots \otimes V^*\).

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