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Questions tagged [linear-algebra]

For questions on linear algebra, including vector spaces, linear transformations, systems of linear equations, spanning sets, bases, dimensions and vector subspaces.

Linear algebra is concerned with vector spaces and linear transformations between them:

$$(x_1, \dots, x_n)\to a_1x_n+\dots+a_nx_n$$

Concepts include systems of linear equations, bases, dimensions, subspaces, matrices, determinants, kernels, null spaces, column spaces, traces, eigenvalues and eigenvectors, diagonalization and Jordan normal forms.

This is a general tag, most of the subjects included have secondary tags, e.g.

127034 questions
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Prove that determinant complex conjugate is complex conjugate of determinant

Prove that determinant complex conjugate is complex conjugate of determinant. I know that there is an induction proof for this. However, I am wondering if there is a more elegant and simple solution that shows this fact. Thanks.
macho
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4 answers

Are determinants always real?

I've just realized that I'm not sure about the answer to this. Are determinants always real-valued? Determinants can be calculated as the product of eigenvalues. Eigenvalues can be complex-valued. Thus surely determinants can be complex-valued? …
user195150
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Does $\mathbb{R}^\mathbb{R}$ have a basis?

I'm studying linear algebra on my own and saw basis for $\mathbb{R}, \mathbb{R}[X], ...$ but there is no example of $\mathbb{R}^\mathbb{R}$ (even though it is used for many examples). What is a basis for it? Thank you
user12205
17
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2 answers

Symmetric matrix multiplication

Let $A$ and $B$ be symmetric matrices. Prove: $AB=BA$ $AB$ is a symmetric matrix As for 1. due to the axiom $(AB)^T=B^T A^T$ so $AB=BA$ As for 2. I did not find any axiom that can support the claim, but from test I found that it is true for…
gbox
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Change of basis = similarity?

Today, the teacher in my class said that any similarity transform of a matrix is essentially a change of basis. So as a result, we end up with the same transformation, just with respect to a different basis. I did not follow that, could someone…
zvdsgf
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17
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How can I represent an N dimensional line?

How can I represent a straight line (between two points) in a N-dimensional space?
jake
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17
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2 answers

Prove $\det(\mathbf I+\mathbf A^T\mathbf A) = \det (\mathbf I+\mathbf A\mathbf A^T)$

Given a matrix $\mathbf A\in \mathbb R^{m\times n}$ and an identity matrix $\mathbf I$ of appropriate dimensions, how do you prove $\det(\mathbf I+\mathbf A^T\mathbf A) = \det (\mathbf I+\mathbf A\mathbf A^T)$?
Learner
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What does it mean for a matrix to be orthogonally diagonalizable?

I'm a little confused as to when a matrix is orthogonally diagonalizable. I understand that if symmetric, it's always orthogonally diagonalizable, but in what other cases can you orthogonally diagonalize a matrix?
google1254
  • 305
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Transpose of product of matrices

How do you prove the following fact about the transpose of a product of matrices? Also can you give some intuition as to why it is so. $(AB)^T = B^TA^T$
user573063
17
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2 answers

The form of $2 \times 2$ unitary matrices

I've been working through "Groups and Symmetry" (Armstrong) and came across this problem in chapter 9 which I can't figure out. Any hints/help would be greatly appreciated! Show that every $2\times 2$ unitary matrix has the…
Matthew C
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Rigorously proving that a change-of-basis matrix is always invertible

How can we prove that all change-of-basis matrices are invertible? The trivial case when it's a change of basis for $\mathbb{R^{n}}$ is easily demonstratable using, for example, determinants. But I am struggling to rigorously show this for all…
user7961
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4 answers

Finding the shortest distance between two lines

I know how to find the distance between a point and a line, not between two lines. Find the shortest distance between the lines $(-1,1,4) + t(1,1,-1)$ and $(5,3,-3) + s(-2,0,1)$ Any help would be appreciated.
Unknown
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17
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2 answers

Kernel of composition of linear transformations

Let $f : U \to V$ and $g : V \to W$ be linear transformations on the vector spaces $U$, $V$, and $W$. Supposedly, $$ \dim(\ker(g \circ f)) = \dim(\ker(f)) + \dim(\ker(g) \cap \operatorname{im}(f)). $$ How might I go about proving that? (Attempt:)…
Snowball
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Proof that the dimension of a matrix row space is equal to the dimension of its column space

I have the following theorem: Theorem 3.12. Let A be an m n matrix. Then the dimension of its row space is equal to the dimension of its column space. And the following proof is given: Proof. Suppose that $\lbrace v_1,v_2,\dots,v_k\rbrace$ is a…
The Pointer
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How to prove a matrix is nilpotent?

Given an $n\times n$ upper triangular matrix $A$ with zero on main diagonal, show that $A^n = 0$. I did some matrix operation and noticed that the diagonal moves up, ultimately all entries will be zero. Is there a nicer way to do it?
Shannon
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