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Found in: Page 191

### Linear Algebra and its Applications

Book edition 5th
Author(s) David C. Lay, Steven R. Lay and Judi J. McDonald
Pages 483 pages
ISBN 978-03219822384

# If A is a $${\bf{4}} \times {\bf{3}}$$ matrix, what is the largest possible dimension of the row space of A? If A is a $${\bf{3}} \times {\bf{4}}$$ matrix, what is the largest possible dimension of the row space of A? Explain.

If A is a $$4 \times 3$$ matrix, then the largest possible dimension of the row space of A is 3.

If A is a $$3 \times 4$$ matrix, then the largest possible dimension of the row space of A is 3.

See the step by step solution

## Step 1: Use the rank theorem

Note that the number of pivots in A gives the dimension of its column space.

By the rank theorem, $$\dim {\rm{Col}}\,A = \dim {\rm{Row}}\,A = {\rm{rank}}\,A$$.

## Step 2: Compute the dimension of the row space for $${\bf{4}} \times {\bf{3}}$$ matrix

If A is a $$4 \times 3$$ matrix, then the number of pivots cannot exceed the number of columns. Here, the number of columns is minimum. Hence, the largest possible dimension of the row space of A is 3.

## Step 3: Compute the dimension of the row space for $${\bf{3}} \times {\bf{4}}$$ matrix

If A is a $$3 \times 4$$ matrix, then the number of pivots cannot exceed the number of rows. Here, the number of rows is minimum. Hence, the largest possible dimension of the row space of A is 3.