Math  /  Algebra

QuestionAnswer the following statements by true or false: T..) Any subset of a vector space that does not contain the zero vector is not a space. ...) The set S={fC[1,1]:f(0)=1}S=\{f \in C[-1,1]: f(0)=1\} is not a subspace of V=C[1,1]V=C[-1,1] ...) S={AR2×2:a11=0}S=\left\{A \in R^{2 \times 2}: a_{11}=0\right\} is a subspace of V=R2×2V=R^{2 \times 2} ...) S={v=(x,y)R2:x+y=1}S=\left\{v=(x, y) \in R^{2}: x+y=1\right\} is not a subspace of V=R2V=R^{2} ...) S={fC(R):f(1)=0}S=\{f \in C(R): f(1)=0\} is a subspace of V=C(R)V=C(R) ...) S={v=(x,y)R2:x+y=1}S=\left\{v=(x, y) \in R^{2}: x+y=1\right\} is not a subspace of V=R2V=R^{2}. )\ldots) is a subspace of S={v=(x,y)R2:x+y=0},V=R2S=\left\{v=(x, y) \in R^{2}: x+y=0\right\}, V=R^{2}

Studdy Solution

STEP 1

1. A subset of a vector space is a subspace if it is closed under vector addition and scalar multiplication, and contains the zero vector.
2. We will evaluate each statement based on these criteria.

STEP 2

1. Evaluate the statement about subsets not containing the zero vector.
2. Evaluate the statement about the set S={fC[1,1]:f(0)=1} S = \{f \in C[-1,1]: f(0)=1\} .
3. Evaluate the statement about the set S={AR2×2:a11=0} S = \{A \in R^{2 \times 2}: a_{11}=0\} .
4. Evaluate the statement about the set S={v=(x,y)R2:x+y=1} S = \{v=(x, y) \in R^{2}: x+y=1\} .
5. Evaluate the statement about the set S={fC(R):f(1)=0} S = \{f \in C(R): f(1)=0\} .
6. Evaluate the statement about the set S={v=(x,y)R2:x+y=1} S = \{v=(x, y) \in R^{2}: x+y=1\} .
7. Evaluate the statement about the set S={v=(x,y)R2:x+y=0} S = \{v=(x, y) \in R^{2}: x+y=0\} .

STEP 3

Any subset of a vector space that does not contain the zero vector is not a subspace. This statement is true because the zero vector is required for a set to be a subspace.

STEP 4

The set S={fC[1,1]:f(0)=1} S = \{f \in C[-1,1]: f(0)=1\} is not a subspace of V=C[1,1] V = C[-1,1] because it does not contain the zero function (since f(0)=1 f(0) = 1 for all fS f \in S ). Therefore, this statement is true.

STEP 5

The set S={AR2×2:a11=0} S = \{A \in R^{2 \times 2}: a_{11}=0\} is a subspace of V=R2×2 V = R^{2 \times 2} . This is because it is closed under addition and scalar multiplication, and it contains the zero matrix. Therefore, this statement is true.

STEP 6

The set S={v=(x,y)R2:x+y=1} S = \{v=(x, y) \in R^{2}: x+y=1\} is not a subspace of V=R2 V = R^{2} because it does not contain the zero vector (since x+y=1 x+y=1 for all vS v \in S ). Therefore, this statement is true.

STEP 7

The set S={fC(R):f(1)=0} S = \{f \in C(R): f(1)=0\} is a subspace of V=C(R) V = C(R) because it is closed under addition and scalar multiplication, and it contains the zero function. Therefore, this statement is true.

STEP 8

This statement is a repetition of STEP_4. Therefore, the statement S={v=(x,y)R2:x+y=1} S = \{v=(x, y) \in R^{2}: x+y=1\} is not a subspace of V=R2 V = R^{2} is true.

STEP 9

The set S={v=(x,y)R2:x+y=0} S = \{v=(x, y) \in R^{2}: x+y=0\} is a subspace of V=R2 V = R^{2} because it is closed under addition and scalar multiplication, and it contains the zero vector. Therefore, this statement is true.
The answers to the statements are:
1. True
2. True
3. True
4. True
5. True
6. True
7. True

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord