a5=0 and a15=4 what is the sum of the first 15 terms of that arithmetic sequence

Usman Zahid
2022-07-24

a5=0 and a15=4 what is the sum of the first 15 terms of that arithmetic sequence

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asked 2021-06-13

For the matrix A below, find a nonzero vector in the null space of A and a nonzero vector in the column space of A

$A=\left[\begin{array}{cccc}2& 3& 5& -9\\ -8& -9& -11& 21\\ 4& -3& -17& 27\end{array}\right]$

Find a vector in the null space of A that is not the zero vector

$A=\left[\begin{array}{c}-3\\ 2\\ 0\\ 1\end{array}\right]$

asked 2021-09-13

Suppose that A is row equivalent to B. Find bases for the null space of A and the column space of A.

$A=\left[\begin{array}{ccccc}1& 2& -5& 11& -3\\ 2& 4& -5& 15& 2\\ 1& 2& 0& 4& 5\\ 3& 6& -5& 19& -2\end{array}\right]$

$B=\left[\begin{array}{ccccc}1& 2& 0& 4& 5\\ 0& 0& 5& -7& 8\\ 0& 0& 0& 0& -9\\ 0& 0& 0& 0& 0\end{array}\right]$

asked 2021-09-18

I need to find a unique description of Nul A, namely by listing the vectors that measure the null space.

$A=\left[\begin{array}{ccccc}1& 5& -4& -3& 1\\ 0& 1& -2& 1& 0\\ 0& 0& 0& 0& 0\end{array}\right]$

asked 2021-12-17

Find k such that the following matrix M is singular.

$\left[\begin{array}{ccc}-3& -3& 3\\ 6& 7& -6\\ 11+k& 16& -15\end{array}\right]$

asked 2022-06-26

Let $V$ be inner product space.

Let ${e}_{1},...,{e}_{n}$ an orthonormal basis for $V$

Let ${z}_{1},...,{z}_{n}$ an orthonormal basis for $V$

I have to show that the matrix represents the transformation matrix between ${e}_{1},...,{e}_{n}$ to ${z}_{1},...,{z}_{n}$ is unitary.

Let ${e}_{1},...,{e}_{n}$ an orthonormal basis for $V$

Let ${z}_{1},...,{z}_{n}$ an orthonormal basis for $V$

I have to show that the matrix represents the transformation matrix between ${e}_{1},...,{e}_{n}$ to ${z}_{1},...,{z}_{n}$ is unitary.

asked 2022-07-08

Consider the 4-by-4 matrix $\mathit{M}={\mathit{M}}_{0}+{\mathit{M}}_{1}$, where

${\mathit{M}}_{0}=\alpha \left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& -1& 0\\ 0& 0& 0& -1\end{array}\right)$ and ${\mathit{M}}_{1}=\beta \left(\begin{array}{cccc}0& \gamma & 0& -{\gamma}^{\ast}\\ {\gamma}^{\ast}& 0& -{\gamma}^{\ast}& 0\\ 0& \gamma & 0& -{\gamma}^{\ast}\\ \gamma & 0& -\gamma & 0\end{array}\right)$

where $\alpha $ and $\beta $ are constants and $\gamma ={\gamma}_{x}+i{\gamma}_{y}$ is complex.

Is it possible to unitary transform $\mathit{M}$ into block off-diagonal form ${\mathit{M}}_{B}$?

Namely, I want to find a unitary transform $\mathit{U}$ so that I can write down ${\mathit{M}}_{B}=\mathit{U}\mathit{M}{\mathit{U}}^{\ast}$ (here ${\mathit{U}}^{\ast}$ is the conjugate transpose).

Explicitly, the required block off-diagonal matrix is (in general form)

${\mathit{M}}_{B}=\left(\begin{array}{cc}0& \mathit{Q}\\ {\mathit{Q}}^{\ast}& 0\end{array}\right)$ where $\mathit{Q}=\left(\begin{array}{cc}{Q}_{z}& {Q}_{x}-i{Q}_{y}\\ {Q}_{x}+i{Q}_{y}& -{Q}_{z}\end{array}\right)$

Is there a general recipe to find such a unitary transformation matrix $\mathit{U}$ which leads to the block off-diagonal form, $\mathit{M}\to {\mathit{M}}_{B}$?

${\mathit{M}}_{0}=\alpha \left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& -1& 0\\ 0& 0& 0& -1\end{array}\right)$ and ${\mathit{M}}_{1}=\beta \left(\begin{array}{cccc}0& \gamma & 0& -{\gamma}^{\ast}\\ {\gamma}^{\ast}& 0& -{\gamma}^{\ast}& 0\\ 0& \gamma & 0& -{\gamma}^{\ast}\\ \gamma & 0& -\gamma & 0\end{array}\right)$

where $\alpha $ and $\beta $ are constants and $\gamma ={\gamma}_{x}+i{\gamma}_{y}$ is complex.

Is it possible to unitary transform $\mathit{M}$ into block off-diagonal form ${\mathit{M}}_{B}$?

Namely, I want to find a unitary transform $\mathit{U}$ so that I can write down ${\mathit{M}}_{B}=\mathit{U}\mathit{M}{\mathit{U}}^{\ast}$ (here ${\mathit{U}}^{\ast}$ is the conjugate transpose).

Explicitly, the required block off-diagonal matrix is (in general form)

${\mathit{M}}_{B}=\left(\begin{array}{cc}0& \mathit{Q}\\ {\mathit{Q}}^{\ast}& 0\end{array}\right)$ where $\mathit{Q}=\left(\begin{array}{cc}{Q}_{z}& {Q}_{x}-i{Q}_{y}\\ {Q}_{x}+i{Q}_{y}& -{Q}_{z}\end{array}\right)$

Is there a general recipe to find such a unitary transformation matrix $\mathit{U}$ which leads to the block off-diagonal form, $\mathit{M}\to {\mathit{M}}_{B}$?

asked 2022-07-03

Consider the $m\times m$-matrix $B$, which is symmetric and positive definite (full rank). Now this matrix is transformed using another matrix, say $A$, in the following manner: $AB{A}^{T}$. The matrix A is $n\times m$ with $n<m$. Furthermore the constraint $rank(A)<n$ is imposed.

My intuition tells me that $AB{A}^{T}$ must be symmetric and positive semi-definite, but what is the mathematical proof for this? (why exactly does the transformation preserve symmetry and why is it that possibly negative eigenvalues in $A$ still result in the transformation to be PSD? Or is my intuition wrong)?

My intuition tells me that $AB{A}^{T}$ must be symmetric and positive semi-definite, but what is the mathematical proof for this? (why exactly does the transformation preserve symmetry and why is it that possibly negative eigenvalues in $A$ still result in the transformation to be PSD? Or is my intuition wrong)?