Math Problem Statement
Determine the dimensions of Nul A, Col A, and Row A for the given matrix. Aequals left bracket Start 4 By 5 Matrix 1st Row 1st Column 1 2nd Column 4 3rd Column 7 4st Column negative 3 5st Column negative 1 2nd Row 1st Column 0 2nd Column 0 3rd Column 0 4st Column negative 7 5st Column negative 1 3rd Row 1st Column 0 2nd Column 0 3rd Column 0 4st Column 0 5st Column 0 4st Row 1st Column 0 2nd Column 0 3rd Column 0 4st Column 0 5st Column 0 EndMatrix right bracket Question content area bottom Part 1 The dimension of Nul A is enter your response here .
Solution
To determine the dimensions of , , and , we analyze the given matrix:
1 & 4 & 7 & -3 & -1 \\ 0 & 0 & 0 & -7 & -1 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{bmatrix}.$$ ### Step 1: Compute the rank of $$A$$ The **rank** of $$A$$ is the number of pivot columns in its row-reduced form. The pivot positions in the matrix are found by performing Gaussian elimination. After row reduction, the matrix $$A$$ becomes: $$\begin{bmatrix} 1 & 4 & 7 & -3 & -1 \\ 0 & 0 & 0 & -7 & -1 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{bmatrix}.$$ Thus, there are **2 pivot columns** ($$\text{rank}(A) = 2$$). ### Step 2: Apply the rank-nullity theorem The **rank-nullity theorem** states: $$\text{rank}(A) + \text{nullity}(A) = \text{number of columns of } A.$$ Here: $$\text{rank}(A) = 2, \quad \text{number of columns} = 5.$$ So: $$\text{nullity}(A) = 5 - 2 = 3.$$ ### Step 3: Dimensions of $$\text{Nul } A$$, $$\text{Col } A$$, and $$\text{Row } A$$ 1. The dimension of $$\text{Nul } A$$ (nullity) is **3**. 2. The dimension of $$\text{Col } A$$ (rank) is **2**. 3. The dimension of $$\text{Row } A$$ (rank) is also **2**. --- ### Final Answer: - $$\dim(\text{Nul } A) = 3$$, - $$\dim(\text{Col } A) = 2$$, - $$\dim(\text{Row } A) = 2$$. Would you like me to show the steps for row reduction or explain the rank-nullity theorem further? --- ### Related Questions: 1. How do you find the basis for $$\text{Nul } A$$ and $$\text{Col } A$$? 2. What is the geometric interpretation of $$\text{Nul } A$$ and $$\text{Col } A$$? 3. How does the rank-nullity theorem apply to square matrices? 4. Can a matrix have a nullity of zero? Why or why not? 5. How do you verify the row rank equals the column rank for any matrix? ### Tip: The dimension of $$\text{Nul } A$$ is the number of free variables in the solution to $$A\mathbf{x} = \mathbf{0}$$.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Theory
Vector Spaces
Formulas
Rank-Nullity Theorem: rank(A) + nullity(A) = number of columns
Theorems
Rank-Nullity Theorem
Properties of Matrix Row Reduction
Suitable Grade Level
Undergraduate Level (Introductory Linear Algebra)