# Tagged Questions

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### What do double vertical lines mean?

I am reading a paper on computer graphic and having hard time to understand this formula: What is the double vertical lines means? Do they always go with power of 2? If I want to learn further ...
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### Same column space is equivalent to same row space?

If $A$ and $B$ are $n \times n$ matrices that have the same column space, then $A$ and $B$ have the same row space. Can one prove or disprove this? This is my continuation of Same row space is ...
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### Same row space is equivalent to same column space?

If $A$ and $B$ are $n \times n$ matrices that have the same row space, then $A$ and $B$ have the same column space. This is false of course. I could just come up with examples though. Can one prove ...
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### A subset that is closed under multiplication but not addition? [duplicate]

I can't get my head around subspaces despite having studied on them quite a lot. Here goes: The problem statement, all given variables and data Give an example of a non-empty subset U of R^2 such ...
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### Show that U subspace is supplementary to the kernel. How to find values of a b c d using intersection of two matrices.

I already found the kernel to be \begin{pmatrix} -2c&-2d\\c&d \end{pmatrix}. and U is a subspace of a $M_2$ matrix defined by \begin{pmatrix} a&b\\2a&2b \end{pmatrix}. So i have to ...
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### Zeros of quadratic form of vectors

I have a set of vectors defined as $[\mathbf{v}(x)]_n = e^{jn\pi x}; \quad n = 0 ~\text{to}~ (N-1)$ where $\mathbf{v}$ is an $N \times 1$ vector, $j$ is $\sqrt{-1}$, and $-1 \leq x < 1$. For a ...
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### Show that a linear matrix transformation is bijective iff A is invertible.

Suppose a linear transformation $T: M_n(K) \rightarrow M_n(K)$ defined by $T(M) = A M$ for $M \in M_n(K)$. Show that it is bijective IFF $A$ is invertible. I was thinking then that I could show ...
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### How do you find a non zero vector in Linear Algebra?

The question is; The vectors $a_1 = (1, 1, 0)$ and $a_2 = (1, 1, 1)$ span a plane in $\Bbb R^3$. Find the projection matrix P onto the plane, and find a nonzero vector $b$ that is projected to zero. ...
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### Finding the least square fit for 3 parameters in Linear Algebra

I know how to find least square for $y = mx+b$ when we have two parameters. But this question has $3$ parameters, am trying to think of how to approach it but so far no success, I can't find any ...
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### Finding the exponential relation between two 4x4 transition matrices

Im alright with matrices, but this question has dumb-struck me. Suppose I have two known and given $4\times4$ transition matrices, representing transitions in three dimensions with the fourth ...
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### Show that $\lambda_1 = \min \{ Q(u) \mid \|u\| = 1 \}$ and $\lambda_m = \max \{ Q(u) \mid \|u\| = 1 \}$

Let $V$ a vector space over $K$ and $Q(u) = \langle u, Tu \rangle$ a quadratic form. $T$ is a symmetric operator. The eigenvalues of $T$ are sorted by size $\lambda_1 < \dots < \lambda_m$. How ...
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### Is the determinant of a matrix some kind of “integral” of the linear mapping?

A $n \times n$ matrix corresponds to a linear mapping between two $n$-dim vector spaces. The determinant of a matrix gives a scalar, just as the integral of an integrable function gives a scalar. ...
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### Write Matrix $A$ to $A = \sum_{i=1}^{3} \lambda_i P_i$

Let $A$ be a Matrix: $$A = \begin{pmatrix} 1 & 0 & 3i \\ 0 & -3 & 0 \\ -3i & 0 & 1 \end{pmatrix}$$ Now I want to write $A$ as $$A = \sum_{i=1}^{3} \lambda_i P_i$$ I ...
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### Write down a matrix of which only the null space is known?

What is the matrix in which null space are all of the multiples of the vector: $$\vec{v}=\begin{bmatrix}4 \\ 3 \\ 2 \\ 1\end{bmatrix}$$ I suppose there are a lot of solutions, but I don't I am not ...
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### Few basic things unclear to me about inner product spaces and orthonormal basis

Few things unclear to me about inner product spaces: assume V is an inner product space with B orthonormal basis. Why is it true that: $$\langle x,y\rangle = \langle[x]_{B} , [y]_B \rangle{st}$$ ...
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### $T: V \rightarrow V$ a linear transformation such that $T^2 = I$ and $H_1= \{v \in V | T(v) = v\}\$ and $H_2= \{v \in V|T(v) = -v\}\$

Let V a vector space and $T: V \rightarrow V$ a linear transformation such that $T^2 = I$ and $H_1= \{v \in V | T(v) = v\}\$ and $H_2= \{v \in V|T(v) = -v\}\$ then $V = H_1 \bigoplus H_2$ I stuck ...
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### maximum and minimum dimension of the space generated by $\{v_1,v_2,v_3,v_4\}$

I'm confused about this problem. I have four vectors $v_1 = (1,1,1,a), v_2 = (1,2,3,a), v_3= (b,1,0,1), v_4 = (0,b,0,0)$ with $a,b$ real numbers. Determine the maximum and minimum dimension of the ...
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### Left shift operator $L: l^2 \rightarrow l^2$ on the sequence space $l^2$

$$L: l^2 \rightarrow l^2$$ is defined by $$b = (b_1,b_2,...) \mapsto Lb = (b_2,b_3,...)$$. $(Lb)_n = b_{n+1}$ respectively. How can I determine the adjoint endomorphism $L^*$? Kind regards George
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### 2x2 symmetric matrix is a subspace of vector space.

Can you kindly check my proof of the problem and correct if possible. The following $S=\{A\in M_{2,2} | AA^T=A^TA\}$ is a subspace of $V=M_{2,2}$ all real $2\times2$ matrices. My proof: S ...
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### Finding loci of possible points satisfying vector simultaneous equations

I recently had an exam and a question came up which I was only partially able to answer. The question was the following: Let $\vec{a}$, $\vec{b}$ and $\vec{c}$ be constant vectors in ...
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### linearly independent vectors and rows/cols space

Given $n$ vectors, we want to determine if those vectors are linearly independent. One way doing it is writing those vectors as columns of a matrix and row-reduce it. The vectors are linearly ...
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### Relationship between the singular value decomposition (SVD) and the principal component analysis (PCA). A radical result(?)

I was wondering if I could get a mathematical description of the relationship between the singular value decomposition (SVD) and the principal component analysis (PCA). To be more specific I have ...
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### Function for diagonalizing a vector.

I was playing around whith the idea of what operation (function) should I perform (apply) over a vector $\mathbf{a} = (a_1,a_2, \ldots, a_N)^T \in \mathbb{R}^N$ to come up with the following matrix: ...
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### kernel space of linear combination of matrices

Suppose $A$ and $B$ are $N\times N$ matrices so that for every $x$ and $y$, $xA+yB$ has a kernel of dimension at least $2$. Is it necessarily true that $\ker(A)\cap\ker(B)$ has dimension at least ...
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### How to find a basis for a tricky 2x 2 matrices vector space

Consider the vector space of 2 x 2 matrices :\begin{bmatrix}a&b\\0&c\end{bmatrix} such that a and c are rational numbers and b is a real number with rational numbers as the field of this ...
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### A basis for the column space of a real matrix

Let $A$ be a real square matrix, and let its column space be $$\mathrm{col}(A)=\{y\in\mathbb{C}^n:y=Ax\text{ for some } x\in\mathbb{C}^n\}.$$ Under what conditions is $\mathrm{col}(A)$ spanned by ...
let $a= \{a_1, a_2, a_3, a_4\}$ and $b=\{b_1,......,b_4\}$ and $r = \{r_1,...,r_4\}$ Also, $b_1 = 4a_1$ $b_2 = 8a_1 + 7a_2$ $b_3 = 4a_1 + 4a_2 + 4a_3$ $b_4 = 9a_1 + 5a_2 + 8a_3 + 5a_4$ and ...