MA25C02 – Linear Algebra
Part A – Important Questions
Is it possible for a vector u to have two different negatives?
Answer: No.
If u + v = 0 and u + w = 0, then v = w.
Hence, the negative of a vector is unique.
If u + v = 0 and u + w = 0, then v = w.
Hence, the negative of a vector is unique.
Is the zero vector a subspace?
Answer: Yes.
The set {0} contains the zero vector and is closed under:
The set {0} contains the zero vector and is closed under:
- Addition
- Scalar multiplication
Is the set of vectors of the form (a, b, 1) a subspace of R³?
Answer: No.
The zero vector (0,0,0) is not of the form (a,b,1). Hence, it is not a subspace.
The zero vector (0,0,0) is not of the form (a,b,1). Hence, it is not a subspace.
Show that S = {(x, y, 0)} is a subspace of R³.
Answer: Yes.
- Contains zero vector (0,0,0)
- Closed under addition
- Closed under scalar multiplication
Is a line passing through origin a subspace?
Answer: Yes.
A line can be written as {t v}, which:
A line can be written as {t v}, which:
- Contains zero vector
- Closed under addition & scalar multiplication
Is the set of matrices with det(A)=0 a subspace?
Answer: No.
It is not closed under addition. Hence, not a subspace.
It is not closed under addition. Hence, not a subspace.
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