If UT is one-to-one, prove that T is one-to-one. must U also be one-to-one?

Nicontio1
2022-01-04
Answered

Let V, W, and Z be vector spaces, and let $T:V\to W$ and $U:W\to Z$ be linear.

If UT is one-to-one, prove that T is one-to-one. must U also be one-to-one?

If UT is one-to-one, prove that T is one-to-one. must U also be one-to-one?

You can still ask an expert for help

GaceCoect5v

Answered 2022-01-05
Author has **26** answers

Let UT is one to one.

It is needed to prove that T is one is one.

Take$UT\left(x\right)=0$ this implies $x=0$ since UT is injective.

Similarly,

$T\left(x\right)=0$

$UT\left(x\right)=U\left(0\right)$

$UT\left(x\right)=0$

Hence, T is injective and is one to one.

Therefore, T is one to one.

If$UT\left(x\right)=$ 0, U may not be one to one as,

$T:{R}^{3}\to {R}^{2}$ defined by $T(a.b,c)=(a.b)$ .

$T:{R}^{2}\to {R}^{3}$ defined by $T(a.b)=(a,b.0)$ .

This states that U may not be one to one.

It is needed to prove that T is one is one.

Take

Similarly,

Hence, T is injective and is one to one.

Therefore, T is one to one.

If

This states that U may not be one to one.

asked 2021-06-01

Find the vectors T, N, and B at the given point.

$r(t)=<{t}^{2},\frac{2}{3}{t}^{3},t>$ and point $<4,-\frac{16}{3},-2>$

asked 2021-09-22

Find the volume of the parallelepiped with one vertex at the origin and adjacent vertices at

asked 2021-05-14

Find a nonzero vector orthogonal to the plane through the points P, Q, and R. and area of the triangle PQR

Consider the points below

P(1,0,1) , Q(-2,1,4) , R(7,2,7)

a) Find a nonzero vector orthogonal to the plane through the points P,Q and R

b) Find the area of the triangle PQR

Consider the points below

P(1,0,1) , Q(-2,1,4) , R(7,2,7)

a) Find a nonzero vector orthogonal to the plane through the points P,Q and R

b) Find the area of the triangle PQR

asked 2022-01-07

Let V, W, and Z be vector spaces, and let $T:V\to W$ and $U:W\to Z$ be linear.

If U and T are one-to-one and onto, prove that UT is also

If U and T are one-to-one and onto, prove that UT is also

asked 2022-01-24

Let $V=Span\{{f}_{1},{f}_{2},{f}_{3}\}$ , where $f}_{1}=1,{f}_{2}={e}^{x},{f}_{3}=x{e}^{x$

a) Prove that$S=\{{f}_{1},{f}_{2},{f}_{3}\}$ is a basis of V.
b) Find the coordinates of $g=3+(1+2x){e}^{x}$ with respect to S.
c) Is $\{{f}_{1},{f}_{2},{f}_{3}\}$ linearly independent?

a) Prove that

asked 2021-11-20

How to calculate the intersection of two planes ?

These are the planes and the result is gonna be a line in$\mathbb{R}}^{3$ :

$x+2y+z-1=0$

$2x+3y-2z+2=0$

These are the planes and the result is gonna be a line in

asked 2022-01-23

How do you find the inner product and state whether the vectors are perpendicular given $<8,4>\cdot <2,4>$