Linear algebra - Transformation using a Matrix

In summary, we are given a linear map T from R^3 to R^3 and a basis B consisting of three vectors. It is also given that (1,0,0) is in the kernel of T. We are asked to find the values of a, b, and c, and a basis for the image of T. By expressing (1,0,0) in the given basis B, we obtain the vector (-1,1,1). Using this, we can solve for a, b, and c by setting up a system of equations. We find that a=1, b=1, and c=1. Therefore, the matrix [T]_B is given by the values a
  • #1
BitterX
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Homework Statement



let [itex]T:R^3 \rightarrow R^3[/itex]
be a Linear map,
and let [itex]B=\left \{ (1,1,1),(1,1,0),(1,0,1) \right \}[/itex]
be a Basis

and [itex](1,0,0)\in kerT[/itex][itex][T]_B=\begin{pmatrix}
a & 0 & b\\
3 & 2a & 1\\
2c& b & a
\end{pmatrix}[/itex]

a. find a,b,c
b. find a Basis for ImT

Homework Equations


The Attempt at a Solution



as for a.

I think that what we need to do is find a general vector (x,y,z)
and express (1,0,0) thorugh it
if we multiply the matrix by what we got we will have to get (0,0,0) (because (1,0,0) is in the kernel)
but I'm not sure how to express (1,0,0).
I think the vector (1,0,0) in the basis B is -1(1,1,1)+1(1,1,0)+1(1,0,1) and so is
(-x+y+z,0,0)

and for the matrix we have
-a+0+b=0
-3+2a+1=0
-2c+b+a=0

from the second equation:
a= 1
from the first
b=1
from the third
c=1

Am I right?
 
Last edited:
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  • #2
yeah look good calling the natural basis E, (1,0,0) in the B basis is as follows
[tex](1,0,0)^T_E = (-1,1,1)^T_B [/tex]

And as a check in E
[tex]-1\textbf{b}_1+1\textbf{b}_2+1\textbf{b}_3 = -1(1,1,1)^T+1(1,1,0)^T+1(1,0,1)^T=(-1+1+1,-1+1+0,-1+0+1)^T =(1,0,0)^T [/tex]
 

1. What is a matrix in linear algebra?

A matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns. It is used to represent linear transformations and can also be used to solve systems of linear equations.

2. How do you perform a transformation using a matrix?

To perform a transformation using a matrix, you multiply the matrix by a vector representing the point or object being transformed. This results in a new vector representing the transformed point or object.

3. What is the purpose of using matrices in linear algebra?

Matrices are used in linear algebra to represent and perform operations on linear transformations. They can also be used to solve systems of linear equations and perform other mathematical calculations.

4. How does a matrix affect the size and shape of an object in a transformation?

A matrix can affect the size and shape of an object in a transformation by scaling, shearing, rotating, or reflecting the object. These transformations can be achieved by multiplying the matrix by a vector representing the object.

5. Can a transformation using a matrix change the orientation of an object?

Yes, a transformation using a matrix can change the orientation of an object by rotating or reflecting it. This is achieved by using a rotation or reflection matrix to multiply the vector representing the object.

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