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Emeli Hagan
2021-01-31
Answered

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l1koV

Answered 2021-02-01
Author has **100** answers

Let the transformation matrix be

Putting in thhe given EQN.

Which is obtained by rotation through angle tyou can use the following transformation matrix as a formula

Where

Where

Solving we get

Hence the transformation matrix is

asked 2021-06-04

Describe the row-echelon form of an augmented matrix that corresponds to a system of linear equations that has an infinite number of solutions.

asked 2021-12-05

Find a basis for the span of the given vectors.

[0 1 -2 1], [3 1 -1 0], [2 1 5 1]

[0 1 -2 1], [3 1 -1 0], [2 1 5 1]

asked 2022-05-17

I am trying to show that the functions ${t}^{3}$ and $b$ are independent on the whole real line. To do this, I try and prove it by contradiction. So assume that they are dependent. So then there must exists constants $a$ and $b$ such that $a{t}^{3}+b|t{|}^{3}=0$ for all $t\in (-\mathrm{\infty},\mathrm{\infty})$. Now pick two points $x$ and $y$ in this interval and assume without loss of generality that $x<0$, $y\ge 0$. Now form the simultaneous linear equations

$a{y}^{3}+b|y{|}^{3}=0$, viz.

$\left[\begin{array}{cc}{x}^{3}& |x{|}^{3}\\ {y}^{3}& |y{|}^{3}\end{array}\right]\left[\begin{array}{c}a\\ b\end{array}\right]=\left[\begin{array}{c}0\\ 0\end{array}\right]$

Now here's my problem. If I look at the determinant of the coefficient matrix of this system of linear equations, namely ${x}^{3}|y{|}^{3}-{y}^{3}|x{|}^{3}$ and noting that $x<0$ and $y>0$, I have that the determinant is non-zero which implies that the only solution is $a=b=0$, i.e. the functions ${t}^{3}$ and |$|t{|}^{3}$ are linearly independent. However what happens if indeed $y=0$? Then the determinant of the matrix is 0 and I have got a problem.

Is there something that I am not getting from the definition of linear independence?

The definition (I hope I state this correctly) is: If $f$ and $g$ are two functions such that the only solution to $af+bg=0\text{}\mathrm{\forall}\text{}t$ in an interval $I$ is $a=b=0$, then the two functions are linearly independent.

But what happens if my functions pass through the origin, like the above? Then I've just shown that there exists a $t$ in an interval containing zero such that the two functions are zero, viz. I can plug in any $b$ and $a$ such that $af+bg=0$.

$a{y}^{3}+b|y{|}^{3}=0$, viz.

$\left[\begin{array}{cc}{x}^{3}& |x{|}^{3}\\ {y}^{3}& |y{|}^{3}\end{array}\right]\left[\begin{array}{c}a\\ b\end{array}\right]=\left[\begin{array}{c}0\\ 0\end{array}\right]$

Now here's my problem. If I look at the determinant of the coefficient matrix of this system of linear equations, namely ${x}^{3}|y{|}^{3}-{y}^{3}|x{|}^{3}$ and noting that $x<0$ and $y>0$, I have that the determinant is non-zero which implies that the only solution is $a=b=0$, i.e. the functions ${t}^{3}$ and |$|t{|}^{3}$ are linearly independent. However what happens if indeed $y=0$? Then the determinant of the matrix is 0 and I have got a problem.

Is there something that I am not getting from the definition of linear independence?

The definition (I hope I state this correctly) is: If $f$ and $g$ are two functions such that the only solution to $af+bg=0\text{}\mathrm{\forall}\text{}t$ in an interval $I$ is $a=b=0$, then the two functions are linearly independent.

But what happens if my functions pass through the origin, like the above? Then I've just shown that there exists a $t$ in an interval containing zero such that the two functions are zero, viz. I can plug in any $b$ and $a$ such that $af+bg=0$.

asked 2021-11-12

**(P^~Q)^(P→Q) **

**check whether it is in contradiction form or not.**

asked 2021-09-18

List five vectors in Span

asked 2022-06-14

Transformation Matrix of a dot product transformation

Let v be a arbitrary vector in R3 and T(x)=v.x. What is the matrix of the transformation T in terms of the components of v? It seems like trying to figure out the matrix using the equation T(x)=Ax does not work, as the left side is a scalar, and the other side is a matrix. Any idea

Let v be a arbitrary vector in R3 and T(x)=v.x. What is the matrix of the transformation T in terms of the components of v? It seems like trying to figure out the matrix using the equation T(x)=Ax does not work, as the left side is a scalar, and the other side is a matrix. Any idea

asked 2022-06-23

Need to find a matrix transformation that takes a (non-square) matrix and within each of its rows keeps the first nonzero element in that row and zeros out the rest of the entries within that row.

I tried to solve the linear matrix equation AX = B to get the matrix transformation A as A = B X^T (XX^T)^{-1} , where X is the matrix to be transformed as such, and B is the desired output matrix (X with only its first nonzero element in each row) but I look for a more straightforward and possibly more elegant way to do it. I appreciate creative answers!

I tried to solve the linear matrix equation AX = B to get the matrix transformation A as A = B X^T (XX^T)^{-1} , where X is the matrix to be transformed as such, and B is the desired output matrix (X with only its first nonzero element in each row) but I look for a more straightforward and possibly more elegant way to do it. I appreciate creative answers!