### Linear Transformation and Matrix Analysis**Question 1:**Let $ T $ be a linear transformation from $ \mathbb{R}^5 $ to $ \mathbb{R}^5 $, and suppose the null space of the matrix for $ T $ has dimension 4.Do the columns of $ T $ span all of $ \mathbb{R}^5 $?- No- Yes**Question 2:**What is the rank of the matrix?- [Text Field for Answer]**Question 3:**Is the transformation invertible?- Yes- No**Question 4:**What is the dimension of the column space?- [Text Field for Answer][Next Question Button]---### Explanation#### Concepts Involved:1. **Null Space and Rank-Nullity Theorem**: - The **null space** of a matrix is the set of all vectors that get mapped to the zero vector when multiplied by the matrix. - The **rank-nullity theorem** states that for a linear transformation represented by a matrix $ A $: \[ \text{Rank}(A) + \text{Nullity}(A) = \text{Number of columns of } A \]2. **Column Space**: - The **column space** is the span of the columns of the matrix. It indicates all possible linear combinations of the column vectors.3. **Invertibility**: - A matrix (or transformation) is **invertible** if there exists another matrix that, when multiplied with it, yields the identity matrix. This occurs if and only if the matrix is square and has full rank.By understanding these concepts, you can determine properties such as the rank of the matrix, whether the transformation is invertible, and the dimension of the column space using the given information about the null space.

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Let T be a linear transformation from R to IR", and suppose the null space of the matrix for T has dimension 4. Do the columns of T span all of R? ONo O Yes What is the rank of the matrix? Is the transformation invertible? O Yes O No What is the dimension of the column space?

Let T be a linear transformation from R to IR", and suppose the null space of the matrix for T has dimension 4. Do the columns of T span all of R? ONo O Yes What is the rank of the matrix? Is the transformation invertible? O Yes O No What is the dimension of the column space?

Algebra and Trigonometry (6th Edition)

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Author:Robert F. Blitzer

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ChapterP: Prerequisites: Fundamental Concepts Of Algebra

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Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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