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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Proof for the distance between a point and a plane

$N$ is a non-zero vector, $c$ a number, and $Q$ a point. $P_0$ is the point of intersection of the line passing through $Q$, in the direction of $N$, and the plane $X\cdot{N} = c$. Show that for all ...
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Minimization of vector norm in the reals

Let $V(x)$ be some vector whose values depend on $x$. That is, $$V(x) = \begin{pmatrix} v_1(x) \\ v_2(x) \\ \vdots \\v_n(x) \end{pmatrix}$$ How can one solve the following equation in the reals?: $$\...
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1answer
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Solving a linear system with one more unknown than equations

I have a linear system with 1 more unknown than equation. My questions are: If I express this equation in matrix form, what 'result' will a computer spit out when solving it (say, with Maple and the ...
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Very confused with polynomial rings and its basis

I am super confused with this in ideals and basis of it. The question asks me to show that $x$ and $x+x^2, x^2$ are minimal basis of the ideal $k[x]$ the set of polynomials of one variable $x$ over ...
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1answer
35 views

What is the property of the eigenvalues of this matrix?

$A+A^{\dagger} = I$ What is the property of the eigenvalues of this matrix? I think they should be real and $A$ is hermitian. But I cannot prove this.
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Isomorphisms between finite vector spaces

Let $U,V$ and $W$ be finite dimensional real vector spaces. Let $L:U\rightarrow V$ and $M:V\rightarrow W$ be one-to-one linear mappings. If $U$ and $W$ are isomorphic, prove that $V$ is isomorphic to $...
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Problem solving three equations using matrices.

Hi I'm not able to solve this system of equations, I use elementary row operations but it's pretty complicated, will you guys help me? $$-x-5y-5z=2$$ $$4x-5y+4z=19$$ $$x+5y-z=-20$$
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2answers
103 views

The image of a square after linear transformation

For this question consider: $T : R ^2 → R ^2$ given by reflection across the line $y = 3x$ and $S : R ^2 → R ^2$ given by rotation anti-clockwise about the origin by an angle of $π/ 2$ . (a) Draw the ...
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Facets of binary polytopes

I have a problem that seems like it should have a slick, elegant solution but I'm having trouble finding one. I'm working with convex polytopes with vertices that are subsets of $\{-1,1\}^n$. When ...
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2answers
39 views

Row operations, matrices

I have a question concerning matrices that I'm not to sure about how to begin. Mostly I don't understand what the question is asking for. Could someone please explain to me what the question is ...
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4answers
61 views

Help with linear algebra linear transformations

Is there a linear transformation $T : R ^2 → R ^3$ such that $T(1, 1) = (1, 0, 2)$ and $T(2, 3) = (1, −1, 4)$. Justify your answer. I'm not sure what exactly this question is asking for. How would ...
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Determining the immersing $N$-dimensional hyper plane of an $(N-1)$-dimensional Hyperplane and a point

Suppose I have an $n-1$ dimensional hypeperplane given as the set of points $x = \lbrace x_1 x_2 \ldots x_n \rbrace, x \in \Bbb{R}^n$ Such that $$ \begin{pmatrix} c_1 x_1 + c_2x_2 + \cdots+ c_nx_n =...
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2answers
40 views

Solution space of a linear system

Whats an example of a 2x3 matrix $A$, if possible such that the solution space of the linear system $Ax=0$ is $\mathbb R^3$? I know the zero matrix is one, but does there exist a non-zero matrix?
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1answer
43 views

Cofactor expansion to check if matrices is invertible.

I gave question regarding a co-factor expansion question. I understand that an easy way to check if a matrices is invertible is to do co-factor expansion and if $A \ne 0$ then its invertible. I'm ...
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0answers
20 views

Equivalent definition of a coset of some subspace of $V$

I am given that $V$ is a vector space over the field $F$, and $X$ is a nonempty subset of $V$ with the following property. $Y=\{x_1-x_2|x_i\in X\}$ is closed under addition and scalar multiplication ...
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1answer
71 views

Which matrices are conjugate to an integer valued matrix?

If I have a matrix $A \in M_{n\times n}(\mathbb{C})$, when does there exist a change of basis $B \in Gl_n(\mathbb{C})$ so that $BAB^{-1} \in M_{n\times n}(\mathbb{Z})$? Case $n=1$ is obvious (in ...
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1answer
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The span of a vector space with elements as linear combinations of no more than $r$ vectors has $dim V \leq r$

If $V=Span \{ \vec{v}_1, \dots, \vec{v}_n \}$ and if every $\{ v_i \}$ is a linear combination of no more than $r$ vectors in $\{ \vec{v}_1, \dots, \vec{v}_r \}$ excluding $\{ v_i \}$, then $dim V \...
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44 views

Find a point given two other points and an angle

I have the 3D coordinates for three points $A, B, C$. I can find the angle formed between the vectors $\vec {AB}$ and $\vec {BC}$ by using a dot product. However, I want to move point $B$ to $B^{\...
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1answer
43 views

If A has an eigenvector, a, and A = C+D then a is an eigenvector of C and D

Is the following true? If A has an eigenvector $\vec{a}$ and if A = C+D then $\vec{a}$ is an eigenvector of C and D. Furthermore if A$\vec{a}$ = $\lambda$$\vec{a}$ and C$\vec{a}$ = $\gamma$$\vec{a}$ ...
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Find all values of 'a' such that given set is a basis of R4?

This is the given set { [1 2 1 3] , [3 4 1 1] , [7 5 a 6] , [1 4 2 a+1] } where we need to find 'a' such that the set is a basis of R4. These are in matrix form with dimension 4 x 1. I proved these ...
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91 views

Linear Algebra - True or False Questions and their Concepts

I have a couple of true-and-false questions whose concepts I need a little help with. 1) If $A$ and $B$ are invertible $n\times n$ matrices, then $(AB)^{-1} = A^{-1}B^{-1}$. 2) If $A$ is a square ...
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Proving that a subset $B$ is also a basis

I just came across a question in a practice exam that asks: Let $B = \{1, t-1, (t-1)^2\}$ be a subset of $P_2$ (all polynomials of degree less than or equal to 2). Prove that $B$ is also a basis for $...
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Multiplying Matrices and getting all possible solutions

So I have the following exercise (Sorry for the terrible formatting) Mx = B where M= \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6\\ 7 & 8 & 9 \end{bmatrix} B= \begin{bmatrix} ...
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2answers
59 views

What is an example of an invertible matrix, A, that has more than one solution to Ax=b?

I understand that the invertibility theorem tells us that Ax=b has at least one solution for every b in R^n . I'm also aware that Ax=0 will have ONLY the trivial solution. What is an example of an ...
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1answer
31 views

Solution Sets of Homogeneous Systems

I had to prove the following theorem: Suppose that $A\mathbf x=\mathbf b$ is consistent for some given $\mathbf b$, and let $\mathbf p$ be a solution. Then the solution set of $A\mathbf x=\mathbf b$ ...
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1answer
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$\overline{\mathbb{Q}}$ subfield of $\mathbb{C}$. [closed]

Prove that the set $\overline{\mathbb{Q}}$ of all the algebraic numbers in $\mathbb{C}$ is a subfield of $\mathbb{C}$.
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2answers
45 views

If and only if criterion for $\xi$ to be algebraic number. [closed]

Prove that $\xi \in \mathbb{C}$ is an algebraic number if and only if the set $\{1, \xi, \xi^2, \xi^3, \dots\}$ is linearly dependent over $\mathbb{Q}$.
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1answer
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Is this true that the roots of $det(P(x))=0$ are Continuous?

Suppose ${A_i}\in {\mathbb{C}^{n \times n}},(i = 0,1,2....m)$ and ${\rm{P(}}x {\rm{) = }}{{\rm{A}}_m}{x ^m} + .....{A_1}x + {A_0}$ is a matrix polynomial. Is this true that the roots of $det(P(x))=...
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Linear algebra - isomorphisms proof confirmation

I asked a question here before about a certain problem which troubled me, and I want to verify if my proof is correct with the new knowledge I obtained. The problem is as follows: Let $\rho : \...
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29 views

The standard vector space over $\mathbb{C}$ generated by $\mathbb{Z}$

I have a problem for my Algebra class which refers to this structure. Could anyone help me understand what this means, because my interpretation (that this is a vector space over $\mathbb{C}$ with a ...
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Prove that function is convex.

I have to prove that the function is convex: $$ f(\mathbf{U}) = ||\mathbf{x} - \mathbf{U} \mathbf{U}^T\mathbf{x}||_2^2 $$ Where $\mathbf{U}$ is matrix which columns are eigenvectors. So $\mathbf{U}^...
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Why does matrix multiplication represent linear transformation compositions?

I know it's probably a silly question, but I'm trying to figure out why was matrix multiplication (the standard one) defined the way it was defined. I know that it was defined like that so we would ...
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1answer
98 views

Subvector matrix notation

Is there any notation indicating a subvector of a matrix ? I need to know the correct way of showing it in an academic paper. i.e: Let $$ A=\begin{bmatrix} 2& -10 & 0 & 4\\ 5& ...
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What is the point of this computation? (Restriction of the Hessian to the normal bundle.)

This is from Banyaga & Hurtubise, Lectures on Morse Theory: Why do the authors say Therefore, the Hessian induces a symmetric bilinear form... ? Isn't this form just obtained by ...
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1answer
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Linear Algebra , matrices [closed]

Let $A = [a_{ij}]$ be a $3\times 3$ matrix such that $\det(A) =-6$. If matrix $B$ is defined by $$B = \begin{bmatrix}3a_{33} & 3a_{32} & 3a_{31} \\ 2\left(a_{31}+a_{23}\right) & 2\...
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2answers
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Simplifying Straight Line Equation to plot a Graph. [closed]

I have the equation: 2y-3 = x I need to simplify the 2y to y so I can plot a straight line. How would I do this? If I reorganise the equation I get something like: 2y = x + 3 but if I then ...
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1answer
25 views

Basis for series that converge for all complex numbers?

Consider the set of power series that converge on all of $\mathbb C$. Clearly that set forms a vector space. Now I noticed that all functions I know which are in that space (that is, all functions ...
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1answer
36 views

Reduced Row Echelon Form

Find all 4x2 matrices in rref form. Attempt at solution. A= \begin{bmatrix} 1 & 0 \\ 0 & 1 \\ 0 & 0 \\ 0 & 0 \end{bmatrix} \begin{bmatrix} 1 & 0 \\ ...
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1answer
34 views

What does a single eigenvector and eigenvalue for a $2 \times 2$ matrix represent geometrically?

I know that eigenvectors for a matrix $A$ represent lines in the plane that are invariant under the the transformation given by $A$, and the corresponding eigenvalues are the scaling that gets applied ...
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1answer
50 views

Specific matrix has no 2-dimensional invariant subspaces

I have the endomorphism $$ M = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{pmatrix} $$ of a real vector space $V$. Note that this matrix is nilpotent (with $M^3 = ...
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1answer
47 views

equation of a line for a computer program

I am writing a computer program which draws the altitudes of triangles and where they intersect. I can use ${y = mx + c}$ to find the equations of two lines and then solve them using simultaneous ...
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1answer
30 views

Lagrange Method of Quadratic Form the a Billinear Form

In the following question I have to present the bilinear form as sum of squares with Lagrange method. $$q(x_1,x_2,x_3,x_4)=2x_1x_4-6x_2x_3$$ However I don't know how I can do it here since none of ...
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118 views

What does it imply if all the eigenvalues of a matrix are all the same?

What properties does a matrix have if all its eigenvalues are the same? In particular, what happens if all eigenvalues are all equal to 1?
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2answers
130 views

Point normal equation of plane

Im doing an homeassignment in linear algebra! This is the question I'm having problems with! The plane M contains two lines; l1 : (x, y, z) = t(2, −1, 0), t ∈ R, l2 : (x, y, z) = t(0, 1, ...
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1answer
49 views

What can be the value of this determinant? [closed]

$\begin{vmatrix} 0 & 1 & 1 & \dots & 1 \\ 1 & 0 & x & \dots & x \\ 1 & x & 0 & \dots & x \\ \vdots & \vdots& \vdots & \ddots & \vdots\\ ...
3
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2answers
213 views

Inequality - GM, AM, HM and SM means

I've got stuck at this problem : Prove that for any $a > 0$ and any $b > 0$ the following inequality is true: $$ {3} {\left(\frac{a^3}{b^3} + \frac{b^3}{a^3}\right)} \geq \frac{a}{b} +\...
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What is the value of the determinant? [duplicate]

\begin{vmatrix} a & b & b & \dots & b \\ b & a & b & \dots & b \\ b & b & a & \dots & b \\ \vdots & \vdots& \vdots & \ddots & \...
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1answer
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What is a Spanning Set of vectors? Why do we need Spanning Sets? [closed]

What is a Spanning Set of vectors? What is the use of Spanning Sets in the real world? Please, explain without using format mathematical notations.
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2answers
363 views

Dot product of the eigenvectors of symmetric positive definite matrix is?

The value of the dot product of the eigenvectors corresponding to any pair of different eigenvalues of a $4−by−4$ symmetric positive definite matrix is _____ . I try to explain $:$ Suppose $...
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1answer
39 views

Equation of line on the complex plane

I'm currently reading a book titled Complex Numbers and Geometry by Liang-shin Hahn, and I'm having trouble understanding the concept of a line in the complex plane as presented in the book. The ...