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In this worksheet, we will practice finding the image and basis of the kernel of a linear transformation.

Q1:

Suppose that the columns of an 𝑚×𝑚 matrix 𝐴 are linearly independent. Then, which of the following statements is always true?

Q2:

Find a basis for ker(𝐴) , where 𝐴=⎛⎜⎜⎝12321021121443302112⎞⎟⎟⎠.

Q3:

Fill in the blank. A linear map 𝑇∈𝐿(𝑉,𝑊) is , if and only if 𝑇 is injective and surjective.

Q4:

Let 𝑇∈𝐿ℝ be such that 𝑇𝑥𝑦=𝑦−𝑥𝑥𝑦∈ℝ.forall Then, 𝑇 is .

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