Question

Let e₁ = [ 1 ], e₂ = [ 0 ], y₁ = [ 2 ] and, y₂ = [ -1 ]

[ 0 ] [ 1 ] [ 5 ] [ 6 ]
and let T: R₂ → R₂ be a linear transformation that maps e₁ into y₁ and maps e₂ into y₂.
Find the images of
[ 5 ] and [ x₁ ]
[-3 ] [ x₂ ]

Answer 1

To find the images of given vectors [5, -3] and [x1, x2] under a linear transformation T which maps basis vectors to known vectors y1 and y2, express each vector as a linear combination of basis vectors and use the mappings to determine their respective images.

Step-by-step explanation:

The student is asking about a linear transformation from R₂ to R₂. Given that the standard basis vectors e₁ and e₂ are mapped to y₁ and y₂ respectively, we can find the image of any vector in R₂ under the transformation T. Since any vector in R₂ can be expressed as a linear combination of e₁ and e₂, we'll use these mappings to find the images of the given vectors.

To find the image of the vector [5, -3], we express it as a linear combination of e₁ and e₂, which is 5e₁ - 3e₂. Applying the transformation T, we get 5T(e₁) - 3T(e₂) which equals 5y₁ - 3y₂. Putting the values of y₁ and y₂ we find the resulting image vector.

In the same way, for the vector [x₁, x₂], its image under T will be x₁T(e₁) + x₂T(e₂), equaling x₁y₁ + x₂y₂. By substituting the values of y₁ and y₂ in this expression, we obtain the corresponding image as a vector in R₂.