Linear algebra, linear trasformation

In summary, the linear transformation L from R2 to R3, defined by L(x)=x1b1+x2b2+(x1+x2)b3, can be represented by the matrix A with respect to the bases (e1,e2) and (b1,b2,b3) as follows: the first column of A is (1,2,1)T and the second column is (1,1,1)T. The correct computation for L(e2) is b1+b3=(1,1,1)T.
  • #1
Mdhiggenz
327
1

Homework Statement



let b1=(1,1,0)T ;b2=(1 0 1)T; b3=(0 1 1)T

and let L be the linear transformation from R2

into R3 defined by

L(x)=x1b1+x2b2+(x1+x2)b3

Find the matrix A representing L with respect to the bases (e1,e2)
and (b1,b2,b3)

Homework Equations


The Attempt at a Solution



First thing I did was label out my e1 and e2

e1=(1,0)

e2=(0,1)

L(e1)=b1+b3
= (1,2,1)T

L(e2)=b2+b3
=(1,1,1)T

So I would assume my A to be (L(e1),L(e2))T
However that is incorrect.

I'm not sure what I am doing incorrect the book does the same steps, but gets a different answer.
 
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  • #2
Row i, column j of A is ##(Le_j)_i## (i.e. the ith component of ##Le_j## in the given ordered basis). This makes ##Le_1## the first column, but you made it the first row.

Also, you computed ##b_2+b_3## wrong.
 

Related to Linear algebra, linear trasformation

1. What is linear algebra?

Linear algebra is a branch of mathematics that deals with the study of linear equations, matrices, vectors, and their properties. It is used to solve systems of linear equations and to represent and analyze linear transformations.

2. What is a linear transformation?

A linear transformation is a function that maps one vector space to another in a way that preserves the structure of the vector space. In other words, it is a function that preserves addition and scalar multiplication.

3. What are matrices and why are they important in linear algebra?

Matrices are rectangular arrays of numbers or symbols that are used to represent and manipulate linear equations and transformations. They are important in linear algebra because they provide a concise and efficient way to represent and solve systems of linear equations.

4. What is the difference between a vector and a scalar?

A vector is a quantity that has both magnitude and direction, while a scalar is a quantity that has only magnitude. In linear algebra, vectors are represented as columns or rows of numbers, while scalars are simply single numbers. Vectors are often used to represent points or directions in space, while scalars are used to represent constants or coefficients in equations.

5. How is linear algebra used in real-world applications?

Linear algebra has many applications in various fields such as physics, engineering, economics, and computer graphics. It is used to solve systems of linear equations, analyze data, and model real-world phenomena. For example, it is used in computer graphics to transform and manipulate images, and in economics to model supply and demand equations.

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