Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them, including systems of linear equations, bases, dimensions, subspaces, matrices, determinants, traces, eigenvalues and eigenvectors, diagonalization, Jordan forms, etc. For questions specifically ...

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Find an orthogonal basis for W.

Use the standard Euclidean inner product on $\mathrm R^4$. Let $W$ be the subspace of $\mathrm R^4$ spanned by $u_1 = (1, 1, 1, 1),$ $u_2 = (2, 4, 1, 5),$ $u_3 = (1, -5, 4, -8).$ Find an ...
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If $\|X(t)\|\leq M$, does this imply that $det(X(t))$ is bounded?

I am wondering if the following is true: If you are given a matrix $X(t)$ (that depends on the positive real variable $t$) which is bounded (i.e, $\|X(t)\|\leq M$ for all $t$. Can you conclude that ...
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290 views

Prove that Either $T$ Is Diagonalizable or $T$ Is Nilpotent.

(Linear Algebra - Hoffman, Kunze, 2nd Ed., Sec 6.8, Q6) Let $V$ be a finite-dimensional vector space over the field $\mathbb{F}$, and let $T$ be a linear operator on $V$ such that $rank \ (T) = 1$. ...
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42 views

Show that concatenating the bases of the eigenspaces of distinct eigenvalues is diagonal

If $T:V\rightarrow V$ is a linear transformation, prove that if $T$ has distinct eigenvalues $\lambda_1,\ldots,\lambda_m$, and a concatenation of the bases of the eigenspaces $E_1,\ldots,E_m$ is a ...
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Is the following matrix Upper Hessenberg?

Does $$ A = \begin{pmatrix} 1 & 1 \\ -1 & -1 \end{pmatrix}$$ properly satisfy the definition of upper Hessenberg?
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Basis polynomial functions

If I suppose $R \subset F$ and have polynomial functions $p_{k,j} : F \to F$ by $p_{1,0}(x)=(x-2)^3$ $p_{2,0}(x)=(x-1)$ $p_{2,1}(x)=(x-1)(x-2)$ $p_{2,2}(x)=(x-1)(x-2)^2$ and the polynomial ...
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A dimension question related to the restriction to a finite-codimensional subspace

Let $V$ be an infinite-dimensional vector space, $T:V \to V$ a linear operator and $W \subset V$ a subspace with $\operatorname{codim} W < \infty$. If $\dim \operatorname{Coker}(T) < \infty$, do ...
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103 views

Proof that real symmetric negative matrix is negative definite

I have a reasonably simple symmetric $p \times p$ matrix $H$, where the $(j,k)$th element is given by $$h_{j,k} = -\sum_{i=1}^n \frac{a_i}{b_i^2} x_{ij} x_{ik}$$ and we know that all $a_i \geq 0$ ...
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382 views

Kernel of adjoint and orthogonal complement images

Alright, suppose we are given $V$, a finite dimensional inner product space, and a linear map, $T:V \rightarrow V$, with its corresponding adjoint, $T^\star :V \rightarrow V$. I want to show: ...
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426 views

Linear transformation's kernel and image of a polynomial

$T:\mathbb{R}[x]_3 \rightarrow \mathbb{R}[x]_3$ defines the linear transformation $$T(p(x))=p''(x)-xp'(x)+2p(x)$$ Write down the matrix of $T$ with respect to the standard basis of $\mathbb{R}[x]_3$ ...
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54 views

Can anyone explain relationship between “onto” and “columns are independent” ?

I remember reading this statement before. It is as follows. Transformation is onto if and only if columns are linearly independnet Transformation is one-to-one if and only if rows are independent ...
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59 views

Determinant on 3x3 matrix and above

When finding the determinent of a matrix, what is the rationale behind multiplying the entry along the row we are deleting from times the cofactor expansion? Also how does doing cofactor expansion ...
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60 views

If an eigenvalue pair $\lambda_1 \cdot \lambda_2 < 0$ exists, then there is a nonzero vector $\vec{v}$ such that $A\vec{v}$ is orthogonal to $\vec{v}$

Suppose $A$ is an invertible, real symmetric $n\times n$ matrix. Prove if $A$ has at least one eigenvalue pair $\lambda_1, \lambda_2$ such that $\lambda_1\cdot \lambda_2 < 0$, then there exists a ...
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35 views

Finding basis for orthogonal subspace

Find a basis for $S^\perp$ for the subspace $$ S = span\left\{\left[\begin{matrix}1\\1\\-2\end{matrix}\right]\right\} $$ How do I start this question?
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81 views

Vector space theorem proof

Given $V$ a vector space, $\mathbf{u}$ is a vector in $V$ and $c$ is a real scalar then 1) $c\mathbf{0}=\mathbf{0}$ 2) $c\mathbf{u}=\mathbf{0}$ $\rightarrow$ $c=0$ or $\mathbf{u}=\mathbf{0}$ How to ...
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91 views

How to find the algebraic multiplicity given the eigenvalues and eigenspaces?

Let A be a 4x4 matrix with eigenvalues $\lambda$ = 2,3 and eigenspaces $E_{\lambda=2} = \operatorname{span} \left\{ {\begin{bmatrix} 1\\0\\0\\1\end{bmatrix}, \begin{bmatrix} ...
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155 views

Orthonormal basis in a cylindrical coordinate system

So I am supposed to show if these vectors make an orthonormal basis in a cylindrical coordinate system. $\vec e_p=\bigl(\begin{smallmatrix} cos(\theta )\\ sin(\theta )\\0 \end{smallmatrix}\bigr); ...
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315 views

Proving $A^2 = 0$ given $A^5 = 0$ [duplicate]

I have a class question where I must prove $A^2 = 0$ given $A^5 = 0$ with A being a 2x2 matrix. I though that I could simply say that as $A^5 = 0$ then $A^2 \cdot A^3 = 0 \implies A^2 = 0$ as $A^2 = ...
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Length of Orthogonal Vectors

Suppose that $u_1$ and $u_2$ are orthogonal vectors, with $||u_1|| = 2$ and $||u_2|| = 5$. Find $||3u_1 + 4u_2||$ $$$$ Then, $u_1 \cdot u_1 = 4$ and $u_2\cdot u_2 = 25$. And $||3u_1 + 4u_2|| = ...
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61 views

Finding a transformation matrix for polynomials

I have a question from the book which says: Given 2 basis for a vector space $R_n[x]$ (Polynomials) and 2 basis $B=(1,x,x^2...,x^n)\quad B'=(1,1+x,1+x+x^2,......,1+x+x^2+....x^n)$ Edited: What is ...
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24 views

Are all matrices of the following form Hermitian?

If I have a matrix $A$ (and it is square and nonsingular), is $A^* A $ Hermitian? Also, does $A $ have to be nonsingular for this to hold?
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Tensor products- balanced maps versus bilinear

When defining tensor products $M\otimes_R N$ over a commutative ring $R$ one can use a universal property with respect to bilinear maps $M\times N\rightarrow P$. On the other hand, in the general ...
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I am going mixed [duplicate]

Why this is wrong ? $ -1=(-1)^{2/2}=((-1)^{2})^{1/2}=1^{1/2}=1 $ Then $1=-1$
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A question on infinite dimensional vector spaces

Let M be an infinite dimensional vector space and $f_1, f_2, \cdots , f_r \in M^*$ be a set of linear independent vectors with $r \geq 2$. Do there exist $m \in M$ such that $f_1(m)\neq 0$ but ...
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Find a matrix representing a given linear transformation [duplicate]

$T(X) = [\{x_1-x_2+x_3\}, \{0+x_2-x_3\}, \{0+0+0\}]$ is a linear transformation from $\mathbb R^3$ to $\mathbb R^3$. Find a matrix $A$ such that $T(x) = A(x)$ Can anyone point me in the right ...
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158 views

Orthogonal columns imply orthogonal rows?

The original question is: Column Vectors orthogonal implies Row Vectors also orthogonal? A counterexample with zero entries is given in one post. However, my question is whether pairwise orthogonal ...
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181 views

Find a basis for $U+W$ and $U\cap W$

Let $$W = \operatorname{span}([2,1,0,1], [0,0,1,0]) \\V = \operatorname{span}([1,2,1,3], [3,1,-1,4])$$ I need to find a basis and the dimension for $U+V$ and $U\cap V$. For $U+V$ I tried: $$U+V = ...
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Definition of onto for linear transformation

I had a question ask the following: "A linear transformation is onto if and only if the columns of its standard matrix form a generating set for its range." To me that seems true but the answer was ...
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31 views

Show that the set $\{e_j\}_{j=1}^\infty \cup \{e_j+e_{j+1}\}_{j=1}^\infty$ is a frame of $l^2$, and find the frame constants $A$ and $B$.

Let $e_j=(0,\dots,0,1,0,\dots)$ where "$1$" is the $j$-th component of the vector. Show that the set $\{e_j\}_{j=1}^\infty \cup \{e_j+e_{j+1}\}_{j=1}^\infty$ is a frame of $l^2$, and find the frame ...
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Explain proof that any positive definite matrix is invertible

If an $n \times n$ symmetric A is positive definite, then all of its eigenvalues are positive, so $0$ is not an eigenvalue of $A$. Therefore, the system of equations $A\mathbf{x}=\mathbf{0}$ has no ...
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615 views

Kernel Explanation

sorry for asking so many questions lately but our lecturer is doing a terrible job explaining things. Calculate $ker(A)$ given that: $f:\{\mathbb{R}^3→\mathbb{R}^3; r→ A\vec{r}\}$ $A= ...
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What's the standard form of the equation of a line of a slanted parabola?

I have been trying to figure out the general form of a slanted parabola, but what I've gotten doesn't look like it would be accurate:$$(x-h)^2+(y-k)^2=\dfrac{d}{\sqrt{h}}$$Where $(h,k)$ is the focus, ...
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1answer
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The main involution on $ M_{2}(F) $ and it's extension to $ M_{2}(F_{\mathbb{A}}) $.

I'm presently reading through a paper of Shimura's; "Special Values of the Zeta Functions Associated with Hilbert Modular Forms". In the paper he defines $ \iota $ to be the main involution of $ ...
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1answer
147 views

Condition of orthogonal eigenvectors

If a matrix $A$ satifies $A^TA=AA^T$, then its eigenvectors are orthogonal. I have not had a proof for the above statement yet. By the way, by the Singular Value Decomposition, $A=U\Sigma V^T$, ...
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48 views

For an arbitrary symmetric matrix, the relation between the number of eigenvalues and the rank of the matrix?

For an arbitrary symmetric matrix $A\in \mathcal{S}^n$, $n$ symmetric space: what is the relation between the number of eigenvalues and the rank of $A$ ? If we know $rank(A) = r$, what is the ...
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523 views

If the null space contains only the zero vector, the map is one-to-one

How does finding out if the null space has only the zero vector prove one-to-one? One-to-one means that there are distinct images for each distinct vector input. $$\mathbb R^n \to \mathbb R^m$$ ...
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81 views

Direct sum and linear independence

By the definition, if $W$ is a vector space and $U_1,U_2$ are the subspaces of $W$ then the direct sum $W=U_1 \oplus U_2$ holds iff the following is true: $W=U_1+U_2$ and every vector in $W$ has a ...
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1answer
50 views

Eigen values of an operator

Suppose A be a $n\times n$ complex matrix. I have to determine eigenvalues of the linear operator $T$ on $n\times n$ matrices such that $T(B)=AB-BA$ in terms of eigenvalues of $A$. A hint will be ...
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what is the difference between linear transformation and affine transformation?

Recently, I am struglling with the difference between linear transformation and affine transformation. Are they the same ? I found an interesting question on the difference between the functions. But ...
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Help with a homework problem involving $\textbf{H}$-conjugate vectors

My problem is the following: Let $\textbf{H}$ be a symmetric $n\times n$ matrix. Are the following claims true? Why? a) If the vectors $\textbf{d}_1$ and $\textbf{d}_2$ and vectors ...
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Easy system of equations?

Given the system of equations find $ v_3,v_4$ . If you only know the value of $ v_1,v_2$ $p_0+p=p_1$ $p_1+p=p_2$ $p_0+2p=p_3$ $p_3+p=p_4$ $p_1v_1 = p_2v_2 = p_3v_3 = p_4v_4$ Came to the equations when ...
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Let $\varphi:G\rightarrow H$ be a homomorphism, show $\varphi':G\rightarrow\text{im}(\varphi)$ is an epimorphism

Let $\varphi:G\rightarrow H$ be a homomorphism, show $\varphi':G\rightarrow\text{im}(\varphi)$ is an epimorphism Epimorphism is a surjective homorphism. We know that $\text{im}(\varphi)\subseteq H$ ...
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152 views

proving a set V is a vector space (in one of the axioms)

If the set $V$ is defined by the points that go through the origin in $\mathbb{R}^2$ that satisfy the equation $ax+by=0$ then show $V$ is a vector space. Resolution:Proving that $V$ is closed under ...
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Why $V_{ij} = \frac {1}{2}(v_iv_j^T + v_jv_i^T),$ is rank-2 if $i\neq j$?

Can someone help me figure out the following argument ? $V_{ij} = \frac {1}{2}(v_iv_j^T + v_jv_i^T),$ is rank-2 if $i\neq j$ where $v_i,v_j \in \mathbb{R}^n$, $v_i,v_j$ are linearly indepedent. ...
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Linear Algebra - Determinant of linear transformation

So I'm working through sample questions and this came up. Any help would be greatly appreciated. Question Let $V$ be the vector space of all complex-valued polynomials $p(x)$ of degree at most $42$ ...
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Why is this a valid definition of the dot product?

$(\vec{u},\vec{v})=u_1v_1+2u_2v_2+3u_3v_3$ I have never seen this definition before. I am used to the dot product looking something like this: $(\vec{a},\vec{b})=a_1b_1+a_2b_2+a_3b_3$ Where do the ...
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1answer
64 views

Decompose a vector into a linear combination of span{Data} and an orthogonal vector

Given $x_1, \ldots, x_n\in \mathbb{R}^D$. Prove that for all $w\in \mathbb{R}^D$, there exists $\alpha_i\in \mathbb{R}$, $v\in \mathbb{R}^D$, such that $$w = \sum_{i=1}^n\alpha_ix_i+v,\\ \forall ...
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2answers
66 views

$3$-linear functions on $\mathbb{R}^3$

I know that any bilinear function $\Phi$ can be presented in a unique way as a sum $$\Phi = S + A,$$ where $S$ is a symmetric and $A$ is skew-symmetric bilinear functions. Does a similar statement ...
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350 views

Having problem with Rotation and Reflection

Show the following, using matrices, combinations of linear transformations, and trigonometric identities. You must prove these in general – an example is not sufficient. (i) The combination of a ...