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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.

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How to solve systems of linear equations for x, y in terms of a, b, c , d

Let $ad - bc \neq 0$. The system is: $$ ax + by = 1 \\ cx + dy = 2$$ I couldn't get far with elimination. I heard you could do this with row reduction, but i'd like an algebraic answer to this. Using substitution, solving for y in the first equation…
user121947
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Coefficients of a linear combination of eigenfunctions

I'm trying to show that given $\psi(x)=c_0\psi_0(x)+c_1\psi_1(x)$, where all functions are normalized and additionally that $\psi_0$ and $\psi_1$ are eigenfunctions of an arbitrary operator, that $|c_0|^2+|c_1|^2 = 1$. From what I understand, the…
Nick T
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Geometric multiplicity, invertible matrices

Would appreciate help with the following problem: Given an invertible $n \times n$ real matrix $A$ with eigenvalue $t$ of geometric multiplicity at least $2$, determine whether the following statement is true: "There is a real polynomial $Q(x)$ of…
michael
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Why is this matrix not in reduced row echelon form?

See matrix G: Why does the textbook say matrix G is not in reduced row echelon form? The zero row is at the bottom, and the other rows have leading 1s in appropriate locations. Thanks
user2461391
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Is Linear transformations $T_1,T_2 : \mathbb{R}^n\rightarrow \mathbb{R}^n$ Invertible?

Let $T_1$ and $T_2$ be two Linear transformations from $\mathbb{R}^n$ to $\mathbb{R}^n$. Let $\{x_1,x_2,\cdots,x_n\}$ be a basis of $\mathbb{R}^n$. Suppose that $T_1(x_i)\neq 0$ for every $1\leq i\leq n$ and that $x_i\perp Ker (T_2)$ for every…
user87543
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$A=I-\frac{1}{n}J$ where $I$ is an $n\times n $ identity matrix, $J$ is an $n\times n $ matrix with all entries as $1$

Consider $$A=I-\frac{1}{n}J$$ where $I$ is an $n\times n $ idenity matrix, $J$ is an $n\times n $ matrix with all entries as $1$. Which of the following is NOT true? $A^k=A$ for every positive integer. Trace of $A$ is $n-1$ $\text{Rank $(A)$+Rank…
user87543
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Dimension of space spanned by row vectors

Question is to find dimension of spaces spanned by vectors : $$\alpha_1=(1,1,0,1,0,0),\\ \alpha_2=(1,1,0,0,1,0),\\ \alpha_3=(1,1,0,0,0,1),\\ \alpha_4=(1,0,1,1,0,0),\\ \alpha_5=(1,0,1,0,1,0),\\ \alpha_6=(1,0,1,0,0,1).$$ I tried to make it down to row…
user87543
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Find all possible Jordan Canonical forms

I am working through the problem below What are the possible Jordan Canonical forms for a matrix $A \in M_n$ with characteristic polynomial $p_A(t)=(t+3)^4(t-4)^2$? Give reason for your answer. Everything I have found regarding this question has…
Tyler Clark
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How prove this matrix $\sigma$ Characteristic polynomial $f(x)=(x+1)^n(x-5)^{\frac{n(n-1)}{2}}(x+1)^{\frac{n(n-1)}{2}}?$

Question: let $$V=\{A|A\in F^{n\times n}\}$$,define linear transformation $$\sigma:A\mapsto 2A-3A^T$$ show that the matrix $\sigma$ Characteristic polynomial $$f(x)=(x+1)^n(x-5)^{\frac{n(n-1)}{2}}(x+1)^{\frac{n(n-1)}{2}}?$$ show that $\sigma$ can…
user94270
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Transformation Matrix representing $D: P_2 \to P_2$ with respect to the basis $B$.

Let $P_2 =\lbrace a_0 +a_1t+a_2t^2:a_0,a_1,a_2 \in R \rbrace$ be the set of polynomials of degree $2$ or less. The linear mapping $D:P_2\to P_2$ is such that $$D(p(t)) = \frac{d}{dt} (t \cdot p(t))$$for $p(t) \in P_2$. a. Show that $B = \lbrace 1,…
Zhoe
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The smallest power of a matrix which equals the identity

This query is inspired by this previous question. Suppose $A$ is an $n \times n$ matrix whose entries are integers between $-s$ and $s$. Suppose further that $A^k=I$ and moreover $k$ is the smallest positive integer with this property. What sort of…
robinson
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Prove that the operator induced by $T$ on the quotient space $ V/\ker[T-5I]$ has all eigenvalues $=0$

The question is as follows : A linear operator T on a complex vector space V has a characteristic polynomial $x^3(x-5)^2$ and minimal polynomial $x^2(x-5)^2$ . Now, i need to prove that the operator induced by T on the quotient space $ V/\ker[T-5I]$…
MathMan
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$A \in M_n(\mathbb R)$ is symmetric: If one of the entries in its diagonal is positive, prove that it has at least one positive eigenvalue

Let $A \in M_n(\mathbb R)$ be symmetric. Given that one of the entries in its diagonal is positive, prove that it has at least one positive eigenvalue. I didn't come to any conclusion. Thanks guys.
Jozef
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What is the convention for sign of rotations in three space?

When performing rotations in three space, is a rotation from x to z considered positive or negative? How do you determine whether similar rotations in three space are positive or negative?
okarin
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A way to solve matrices within equation

Solve for X $$2X+X^t=B$$ Where $X,B, $are matrices, and $ X^t$ stands for transpose. I was trying to work with indices but it doesn't seems to work...
user7777777
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