Question

Which of the following sets form a basis for the subspace H, i.e., which sets form an efficient spanning set containing no unnecessary vectors?

a) {v1, v2, v3}
b) {v1, v2}
c) {v2, v3}
d) {v1, v3}

Answer 1

To determine which sets form a basis for the subspace H, we need to check if they constitute a spanning set and if they contain no unnecessary vectors. By applying these criteria to each set, we can determine which ones form a basis for the subspace H.

Step-by-step explanation:

To determine which sets form a basis for the subspace H, we need to check if they constitute a spanning set and if they contain no unnecessary vectors. A set is a spanning set if any vector in the subspace H can be expressed as a linear combination of the vectors in the set. To check for unnecessary vectors, we need to ensure that none of the vectors in the set can be expressed as a linear combination of the other vectors. By applying these criteria to each set, we can determine which ones form a basis for the subspace H.

a) {v1, v2, v3}: To check if this set is a spanning set, we need to determine if any vector in the subspace H can be expressed as a linear combination of v1, v2, and v3. If this is the case, then the set is a spanning set. To check if there are unnecessary vectors, we need to determine if any vector in the set can be expressed as a linear combination of the other vectors. If this is true, then the set contains unnecessary vectors. You can go through the calculations to check if these conditions are met for this set.

b) {v1, v2}: To check if this set is a spanning set, we need to determine if any vector in the subspace H can be expressed as a linear combination of v1 and v2. If this is the case, then the set is a spanning set. To check if there are unnecessary vectors, we need to determine if any vector in the set can be expressed as a linear combination of the other vectors. If this is true, then the set contains unnecessary vectors. You can go through the calculations to check if these conditions are met for this set.

c) {v2, v3}: To check if this set is a spanning set and contains no unnecessary vectors, you can follow the process outlined above.

d) {v1, v3}: To check if this set is a spanning set and contains no unnecessary vectors, you can follow the process outlined above.

After applying the criteria to each set, you can determine which ones form a basis for the subspace H.