Math  /  Algebra

QuestionSuppose that AA is an n×mn \times m matrix of rank 4 , the nullity of AA is 2 , and the column space of AA is a subspace of R8\mathbf{R}^{8}. Find the dimensions of AA. AA has \square rows and \square columns.

Studdy Solution

STEP 1

What is this asking? We're given some facts about a matrix AA, specifically its rank, nullity, and the space its columns live in, and we need to figure out how many rows and columns it has. Watch out! Don't mix up the row space and column space!
Also, remember the Rank-Nullity Theorem!

STEP 2

1. Relate Rank and Rows
2. Use the Rank-Nullity Theorem
3. Find the Number of Columns

STEP 3

We're told the column space of AA is a subspace of R8\mathbf{R}^{8}.
This tells us that the columns of AA each have **8 entries**.
Since each column represents a vector in R8\mathbf{R}^{8}, we know that AA has **8 rows**.

STEP 4

The **Rank-Nullity Theorem** says that the rank of a matrix plus its nullity equals the number of columns.
We can write this as: rank(A)+nullity(A)=number of columns of A \text{rank}(A) + \text{nullity}(A) = \text{number of columns of } A

STEP 5

We know that the **rank** of AA is **4** and the **nullity** is **2**.
Let's plug these values into our equation: 4+2=number of columns of A \mathbf{4} + \mathbf{2} = \text{number of columns of } A

STEP 6

From the previous calculation, we can easily see that: number of columns of A=6 \text{number of columns of } A = \mathbf{6} So AA has **6 columns**.

STEP 7

AA has **8** rows and **6** columns.
So the dimensions are 8 x 6.

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