# What is the smallest subspace of 3 by 3 matrices?

## What is the smallest subspace of 3 by 3 matrices?

The trivial substace, consisting of a 3×3 null-matrix, is the smallest subspace of the vector space of all symmetric and lower-triangular 3×3 matrices, since it contains only one element, the 3×3 null-matrix, which satisfies both of your conditions.

## What is the smallest subspace?

spanU is the smallest among subspaces.

How do you find the subspace of a matrix?

Let A be an m × n matrix.

1. The column space of A is the subspace of R m spanned by the columns of A . It is written Col ( A ) .
2. The null space of A is the subspace of R n consisting of all solutions of the homogeneous equation Ax = 0: Nul ( A )= C x in R n E E Ax = 0 D .

### Which of the following would be the smallest subspace containing the first quadrant of the space?

The smallest subspace containing the first quadrant is the whole space R2. If we start from the vector space of 3 by 3 matrices, then one possible subspace is the set of lower triangular matrices.

### What is the smallest subspace of the space of 4×4 matrices which contains all upper triangular matrices?

Therefore, the smallest subspace of the space of 4 × 4 matrices which contains all upper triangular matrices (aj,k = 0 for all j > k), and all symmetric matrices (A = AT ) is the whole space M4×4. For the second part, if a matrix is both upper triangular and symmetric, it must be diagonal.

Is the smallest subspace of V contains?

Hence, the smallest subspace of V containing S is [S].

## What is the subspace of a matrix?

Definition: A Subspace of is any set “H” that contains the zero vector; is closed under vector addition; and is closed under scalar multiplication. Definition: The Column Space of a matrix “A” is the set “Col A “of all linear combinations of the columns of “A”.

## How do you work out the size of a subspace?

Dimension of a subspace As W is a subspace of V, {w1,…,wm} is a linearly independent set in V and its span, which is simply W, is contained in V. Extend this set to {w1,…,wm,u1,…,uk} so that it gives a basis for V. Then m+k=dim(V).

Why span is the smallest subspace?

For minimality, suppose S ว V0, a subspace of V. V0 contains all linear combination of elements of S. That is, span(S) ว V0. Hence, span(S) is the smallest subspace containing S.

### What is the subspace of R3?

Every line through the origin is a subspace of R3 for the same reason that lines through the origin were subspaces of R2. The other subspaces of R3 are the planes pass- ing through the origin. Let W be a plane passing through 0. We need (1) 0 ∈ W, but we have that since we’re only considering planes that contain 0.

### Is it possible for v1 v2 v3 to be a linear combination of v1 and v1 v2 explain why or why not?

No, it is impossible: If the vectors v1,v2,v3 are linearly dependent, then one of the vectors is a linear combination of two others. Therefore the subspace V := span{v1,v2,v3} is generated by these 2 vectors.

How do I find the subspace of R3?

In other words, to test if a set is a subspace of a Vector Space, you only need to check if it closed under addition and scalar multiplication. Easy! ex. Test whether or not the plane 2x + 4y + 3z = 0 is a subspace of R3.

## Can a set of 3 vectors span R2?

In general 1. Any set of vectors in R2 which contains two non colinear vectors will span R2. 2. Any set of vectors in R3 which contains three non coplanar vectors will span R3.

What is a subspace of r3?

A subset of R3 is a subspace if it is closed under addition and scalar multiplication. Besides, a subspace must not be empty. The set S1 is the union of three planes x = 0, y = 0, and z = 0. It is not closed under addition as the following example shows: (1,1,0) + (0,0,1) = (1,1,1).

### What is the largest subspace of a vector space?

There are many possible answers. One possible answer is {x−1,x2−x+2,1}. What is the largest possible dimension of a proper subspace of the vector space of 2×3 matrices with real entries? Since R2×3 has dimension six, the largest possible dimension of a proper subspace is five.