# Tagged Questions

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 ...

46 views

33 views

### Matrix that changes basis

Does change of basis matrix we use in linear transformations change both the domain and range of the transformation matrix? By the way, I have a hard time calculating the change of basis matrix. I've ...
45 views

### Show that the diagonal elements are not all $0$

If the rank of a real symmetric matrix be $1$, show that the diagonal elements of the matrix can not be all zero. Since the rank is $1$, the determinant of the entire matrix is $0$, so it is ...
41 views

### represent an image in linear algebra

Can we represent a grayscale image as a matrix of values, and then apply all our linear algebra techniques to that? Like finding the column space and null space, reason about the matrix structure. ...
29 views

### Finding Curvatures of Parabolas

Use the formula for the center of curvature c of a curve $r=r(t)$ at a arbitrary point r(t) given by $c-r=-\dot r^{3}(\dot r \land \ddot r)^{-1}$ to find the center of curvature of the semicubical ...
45 views

### When do $k$ vectors span $\mathbb{R}^n$? ($k>n$)

My specific question I'm having trouble with is finding the values of $a$ for which $v_1=(1,3,4), v_2=(2,-1,1), v_3=(-3,5,a^2-2), v_4=(4,2,a+4)$ span $\mathbb{R}^3$. I'm relatively new to Linear ...
18 views

### Number of Subspaces that contains other Space

In $GF(2)$, How Can I calculate the number of subspaces of dimension $k<w$ that contains a fixed subspace of dimension $k'<w$:
40 views

### Intuitive way to understand the use of matrix inversion to find dual basis

I'm currently thinking about the following problem: Problem: Let $B = (b_1, b_2, b_3)$ a base of $\mathbb{R}^3$. Find the correlating dual basis $B^* = (b_1^*, b_2^*, b_3^*)$. $B$ is explicitly ...
23 views

32 views

12 views

46 views

### How do I write $B = \left\{\left[\begin{smallmatrix} x \\ y \end{smallmatrix}\right] \in \boxed{?}| \ldots\right\}$ with proper notation

Let $x \in X \subset \mathbb{R}^n$, then I define a set: $$A = \{x \in X| 1^Tx = 0\}$$ Now supose I have another element $y \in Y \subset \mathbb{R}^n_{+}$ I concatenate $x,y$ in to a single vector ...
21 views

### I need help for steps in how to solve for $L$ [on hold]

$$-(1-L)^{-\frac{1}{2}}L^{\frac{1}{2}} + (1-L)^{\frac{1}{2}}L^{-\frac{1}{2}}=0$$ Thanks in advance, I've been stuck on this for a while. Chris
25 views

### What does a function of matrices do to the eigenvalues of matrices in its domain? Two examples and request for generalization if possible

I think, for example, that if $\lambda$ is an eigenvalue of a matrix $A$, then $\lambda^2$ is an eigenvalue for $A^2$ and that $\frac{1}{\lambda}$ is an eigenvalue for $A^{-1}$ provided $A$ is ...
78 views

18 views

29 views

### Determining a basis for Col($A$) and a dimension for the null space of $A$

Let $A = \begin{bmatrix}1&-1&1&0&-2&1\\1&-1&1&1&0&0\\-1&1&-1&2&5&-1\end{bmatrix}$ a) Determine a basis for Col($A$) b) What is the ...
41 views

### Lower bounding the trace of $A^2$ using the trace of $A^T A$

$\DeclareMathOperator{\tr}{tr}$For a real, square matrix $A$, I believe that one has a simple upper bound on the (absolute value of the) trace of its square in terms of the trace of its Gramian-type ...
34 views

### What if 1st pivot is missing but the 2nd one is there?

I have the following matrix : $$A= \begin{bmatrix} 0 &1 &2 &3 &4 \\ 0 &0 &0 &1 &2\\ 0 &0 &0 &0 &0\\ \end{bmatrix}$$ So here the 1st pivot is missing ...
Let $A = \begin{bmatrix}4&-4&2&-6\\2&-2&1&-3\end{bmatrix}$ Find a basis of nullspace$(A)$ I first put $A$ in RREF to get: \$\begin{bmatrix}1&-1&1/2&-3/2\\0&0&...