BolkowN

2020-11-05

Let $T\text{}:\text{}U\text{}\to \text{}U$ be a linear transformatiom and let

$\mathcal{B}$ be a basis of U. Define the determanant det (T) of T as

$det(T)=det([T]\text{}\mathcal{B}).$

Show that det (T) is well=defined, i.e. that it does not depend on the choice of the basis$\mathcal{B}.$

Prove that T is invertible if and only if det$(T)\ne \text{}0.$ If T is invertible, show that

$det({T}^{-1})=\text{}\frac{1}{det(T)}.$

Show that det (T) is well=defined, i.e. that it does not depend on the choice of the basis

Prove that T is invertible if and only if det

coffentw

Skilled2020-11-06Added 103 answers

To establish some basic facts about the determinant of a linear transformation T from U to U (U a vector space)

A linear transformation

Let B be a basis of U. Then T is represented as a square matrix of size n (n= dimension of U)

Let

(The vectors in B are linearly independent and they span U).

Then any linear map

With the above notation, we define det (T) = determinant of the matrix

In order that det (T) is well-defined , we need to show that this definition does not depend on the choice of the basis B. In other words, if C is some other basis for U, we need to prove that

Let B, C are rwo bases of U.

We need to show

which in turn proves that det(T) is well - defined.

(det(T) is independent of the basis)

Here , we have used the standard properties of determinants of matrices.

Proof: Let B, C are two bases of U.

Then

where P is the change of basis matrix.

Now

Hence, the proof

Proof of T is invertible iff det (T) is not zero

Now,

What is the area of the parallelogram with vertices A(-3, 0), B(-1, 5), C(7, 4), and D(5, -1)?

How to expand and simplify $2(3x+4)-3(4x-5)$?

Find an equation equivalent to ${x}^{2}-{y}^{2}=4$ in polar coordinates.

How to graph $r=5\mathrm{sin}\theta$?

How to find the length of a curve in calculus?

When two straight lines are parallel their slopes are equal.

A)True;

B)FalseIntegration of 1/sinx-sin2x dx

Converting percentage into a decimal. $8.5\%$

Arrange the following in the correct order of increasing density.

Air

Oil

Water

BrickWhat is the exact length of the spiraling polar curve $r=5{e}^{2\theta}$ from 0 to $2\pi$?

What is $\frac{\sqrt{7}}{\sqrt{11}}$ in simplest radical form?

What is the slope of the tangent line of $r=-2\mathrm{sin}\left(3\theta \right)-12\mathrm{cos}\left(\frac{\theta}{2}\right)$ at $\theta =\frac{-\pi}{3}$?

How many integers from 0 to 50, inclusive, have a remainder of 1 when divided by 3?

Use the summation formulas to rewrite the expression $\Sigma \frac{2i+1}{{n}^{2}}$ as i=1 to n without the summation notation and then use the result to find the sum for n=10, 100, 1000, and 10000.

How to calculate the right hand and left hand riemann sum using 4 sub intervals of f(x)= 3x on the interval [1,5]?