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scholesmu
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Q1:
Let A =
1 3 0
0 -4 4
3 4 5
a) Find Cartesian equations for Col(A) and Null(A).
b) Are the columns of A linearly independent? Give full reasons for your answer (based on
the definition of linear independence).
Q2: Prove that in any vector space k × 0 = 0, where k is any real number and 0 is the zero vector.
Q3:
The set M2,2 of 2 × 2 matrices, with real entries, is a vector space.
The set of diagonal matrices D =
[(a 0
0 b) | a, b 2 E R is a subset of M2,2
a) Write down two particular matrices which belong to D, and two particular matrices which
belong to M2,2 but not to D.
b) Prove that D is a subspace of M2,2
Let A =
1 3 0
0 -4 4
3 4 5
a) Find Cartesian equations for Col(A) and Null(A).
b) Are the columns of A linearly independent? Give full reasons for your answer (based on
the definition of linear independence).
Q2: Prove that in any vector space k × 0 = 0, where k is any real number and 0 is the zero vector.
Q3:
The set M2,2 of 2 × 2 matrices, with real entries, is a vector space.
The set of diagonal matrices D =
[(a 0
0 b) | a, b 2 E R is a subset of M2,2
a) Write down two particular matrices which belong to D, and two particular matrices which
belong to M2,2 but not to D.
b) Prove that D is a subspace of M2,2