Math  /  Algebra

Question34. Let VV and WW be two subspaces of Rn\mathbb{R}^{n}. (a) Is VWV \cap W a subspace of Rn\mathbb{R}^{n} ? (b) Is VWV \cup W a subspace of Rn\mathbb{R}^{n} ?

Studdy Solution

STEP 1

What is this asking? We're checking if the intersection and union of two subspaces are also subspaces. Watch out! Remember the key properties of a subspace: it must contain the zero vector, be closed under addition, and be closed under scalar multiplication.
Don't mix up intersection and union!

STEP 2

1. Intersection of Subspaces
2. Union of Subspaces

STEP 3

Let's **tackle** the intersection VWV \cap W first!
This is the set of all vectors that are in *both* VV and WW.

STEP 4

Since VV and WW are subspaces, they both contain the **zero vector**, 0\mathbf{0}.
If something is in *both* VV and WW, it's definitely in their intersection!
So, 0\mathbf{0} is in VWV \cap W. **Awesome!**

STEP 5

Let's pick two vectors, u\mathbf{u} and v\mathbf{v}, that are *both* in VWV \cap W.
This means u\mathbf{u} and v\mathbf{v} are in VV, *and* u\mathbf{u} and v\mathbf{v} are in WW.
Since VV is a subspace, u+v\mathbf{u} + \mathbf{v} is in VV.
Similarly, since WW is a subspace, u+v\mathbf{u} + \mathbf{v} is in WW.
If it's in *both*, then u+v\mathbf{u} + \mathbf{v} is in VWV \cap W. **Boom!** Closure under addition **confirmed!**

STEP 6

Let's grab a scalar cc and a vector u\mathbf{u} in VWV \cap W.
So, u\mathbf{u} is in both VV and WW.
Since VV is a subspace, cuc\mathbf{u} is in VV.
Since WW is a subspace, cuc\mathbf{u} is in WW.
Therefore, cuc\mathbf{u} is in VWV \cap W. **Nailed it!** Closure under scalar multiplication: **check!**

STEP 7

Now, let's **explore** the union VWV \cup W.
This set contains all vectors that are in *either* VV or WW (or both).

STEP 8

Let's think about R2\mathbb{R}^2 (the 2D plane).
Imagine VV is the x-axis, and WW is the y-axis.
Both are subspaces.
The vector (10)\begin{pmatrix} 1 \\ 0 \end{pmatrix} is in VV, and the vector (01)\begin{pmatrix} 0 \\ 1 \end{pmatrix} is in WW.
So, both vectors are in the union VWV \cup W.

STEP 9

Let's add them: (10)+(01)=(11)\begin{pmatrix} 1 \\ 0 \end{pmatrix} + \begin{pmatrix} 0 \\ 1 \end{pmatrix} = \begin{pmatrix} 1 \\ 1 \end{pmatrix}.
Is this new vector in VWV \cup W? **Nope!** It's not on either the x-axis or the y-axis.
So, VWV \cup W is *not* closed under addition.

STEP 10

(a) Yes, VWV \cap W is a subspace of Rn\mathbb{R}^{n}. (b) No, VWV \cup W is not necessarily a subspace of Rn\mathbb{R}^{n}.

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