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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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Orthogonal complement of orthogonal complement

Let $U$ be a subspace of $V$ (where $V$ is a vector space over $C$ or $R$). The orthogonal complement of the orthogonal complement of $U$ is not equal to $U$ in general (equal only for dim $V$ finite). Can anyone give me a simple example when the…
John Michael
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Why do elementary steps of elimination appear to fail for a system of equations with each term of degree 2?

I apologise in advance if this is irritating due to its simplicity - I have only done High School Mathematics. Given: (1) $(a+1)^2$ = $(b+2)^2$+$(c+3)^2$ (2) $(a+3)^2$ = $(b+1)^2$+$(c+2)^2$ (3) $(a+4)^2$ = $(b+3)^2$+$(c+1)^2$ (4) $(a+2)^2$ =…
David
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Natural inner product on the Hom space.

Let $P$ and $V$ be vector spaces of dimension $k$ and $n$, respectively. I want to know if it's possible to endow $\text{Hom}(P,V)$ with a natural inner product, but without using the (non-canonical) isomorphism between $V$ and $V^*$ and then…
Bias of Priene
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A linear map of the plane which preserves linear independence over Q but not over R

This is directly from Stewart & Tall's Algebraic Number Theory & FLT, chapter 8, Exercise 6, page 150. I can construct maps which do weird things but haven't been able to do this. Matrix representations haven't paid off YET. This map obviously must…
David Dyer
  • 481
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Operators such that $T\circ S=I$ but $S\circ T\neq I$.

Suppose $S$ and $T$ are linear operators on a vector space $V$ and $T\circ S=I$ where $I$ is the identity map. It's easy to see that $S$ is one-to-one. If $V$ is finite dimensional, rank-nullity implies $S$ is invertible, so a little manipulation…
Kally
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When does $Ax \geq Ay $ imply $x \geq y$?

Under what conditions on $A$ does $Ax \geq Ay $ imply $x \geq y$? In the context of a larger proof, I have an upper triangular matrix $A$ and two nonzero vectors $x$ and $y$. I have shown that $Ax \geq Ay $, and it seems that this implies $x \geq y$…
Max
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Why is the "non-zero vector" hypothesis necessary here?

I'm self-studying linear algebra and don't want to post too many details here in case someone is using the same textbook. I ran across this simple exercise where I think there is an extra hypothesis. To prove: Suppose $ A \in \mathbb{R}^n, A \neq 0,…
Steve
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use linearly independence prove

Let $A\in{\mathbb{R}^{m\times n}}$. Suppose $x_{1},\ldots,x_{k}$ are vectors in ${\mathbb{R}^{n}}$ and {${Ax_{1},\ldots,Ax_{k}}$} is a linearly independent set. this problem has three problem. (a)Prove that{$x_{1},\ldots,x_{k}$} is linearly…
81235
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Can a "dimension reducing" function be bijective?

Can a function whose image has lower dimensionality than its domain (alas I don't know if there is a special name for that kind of functions) ever be bijective? Consider e. g. $$f:X\to Y:\begin{pmatrix} x_1\\x_2\\x_3 \end{pmatrix} \to\begin{pmatrix}…
chicken_game
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Find a vector $y$ such that $g(x)=\langle x,y \rangle$ for all $x \in V$

$V$ is an inner product space and $g: V \rightarrow F$ is a linear transformation. Find a vector $y$ such that $g(x)=\langle x,y \rangle$ for all $x \in V$ $V=P_2(R)$ with $\langle f,h \rangle=\int_{0}^{1}f(t)h(t)dt$, $g(f)=f(0)+f'(1)$. I know, by…
noname
  • 253
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$A \in M_{n \times n} (\mathbb R)$, $n\geq 2$, rank($A) = 1$, trace($A) = 0$. Prove A is not diagonalizable

Given: $A \in M_{n \times n} (\mathbb R)$, $n\geq 2$, rank($A) = 1$, trace($A) = 0$. Prove A is not diagonalizable and find $P_A(x)$. So I said: if $n \geq 2$ and rank($A)=1$ then $A$ is not invertible. That means that it has an eigenvalue 0.…
TheNotMe
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$A,B$ are complex matrices in $M(n,\mathbb{C})$ and rank$(AB)$=rank$(BA)$=rank$(ABA)$. Prove that $AB$ is similar to $BA$.

$A,B$ are complex matrices in $M(n,\mathbb{C})$ and rank$(AB)$=rank$(BA)$=rank$(ABA)$. Prove that $AB$ is similar to $BA$. I tried by using JCF..but could not complete
user898465
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matrices such that $A^2B+BA^2=2ABA$

Let be $A$, $B$ two matrices $3 \times 3$ with complex entries. Prove that if $$A^2B+BA^2=2ABA$$ THEN $$B^2A+AB^2=2BAB$$ I tried it and do not know how to continue. If $A$ is invertible then $$AB^2+A^{-1}BA^2B=2BAB$$ so I have to prove…
shangq_tou
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A theorem in Linear Algebra; linear dependence - Axler

I really am having trouble understanding the statement and the proof. Why does the theorem pick $v_1 \neq 0 $? Why not $v_2$? Also in proving (a), why do we consider the largest $j$? I do not understand the statement Note all of $a_2, a_3, \dots,…
Lemon
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Finding dim null T for nilpotent operators

I was working through some problems when I came across this one that I was having some trouble with. Assume that $\dim V = n$. Assume that $n > 20130520$. Let $T$ be an operator on $V$. (1) If $T^{n-1} \neq 0$, $T^n = 0$. Find $\dim…
Rob Jordan
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