(a) If X or Y is linearly dependent, it means that there exists a non-trivial linear combination of vectors in X or Y that equals the zero vector. Let's assume without loss of generality that X is the linearly dependent set. This means that there exist scalars a1, a2, ..., an, not all zero, and vectors x1, x2, ..., xn in X such that:

a1*x1 + a2*x2 + ... + an*xn = 0

Now, consider the set X ∪ Y. This set contains all vectors in X and Y. Therefore, the above non-trivial linear combination of vectors in X is also a non-trivial linear combination of vectors in X ∪ Y. Hence, X ∪ Y is linearly dependent.

(b) Consider the vector space R^3 over the field of real numbers. Let X = {(1, 0, 0)} and Y = {(0, 1, 0), (0, 0, 1)}. Both X and Y are linearly independent sets because no vector in either set can be written as a linear combination of the others. However, X ∪ Y = {(1, 0, 0), (0, 1, 0), (0, 0, 1)} is linearly dependent because the vector (1, 0, 0) can be written as a linear combination of the vectors (0, 1, 0) and (0, 0, 1).

Question

(a) If X or Y is linearly dependent, it means that there exists a non-trivial linear combination of vectors in X or Y that equals the zero vector. Let's assume without loss of generality that X is the linearly dependent set. This means that there exist scalars a1, a2, ..., an, not all zero, and vectors x1, x2, ..., xn in X such that:

a1*x1 + a2*x2 + ... + an*xn = 0

Now, consider the set X ∪ Y. This set contains all vectors in X and Y. Therefore, the above non-trivial linear combination of vectors in X is also a non-trivial linear combination of vectors in X ∪ Y. Hence, X ∪ Y is linearly dependent.

(b) Consider the vector space R^3 over the field of real numbers. Let X = {(1, 0, 0)} and Y = {(0, 1, 0), (0, 0, 1)}. Both X and Y are linearly independent sets because no vector in either set can be written as a linear combination of the others. However, X ∪ Y = {(1, 0, 0), (0, 1, 0), (0, 0, 1)} is linearly dependent because the vector (1, 0, 0) can be written as a linear combination of the vectors (0, 1, 0) and (0, 0, 1).

Knowee AI · Accepted Answer

(a) If X or Y is linearly dependent, it means that there exists a non-trivial linear combination of vectors in X or Y that equals the zero vector. Let's assume without loss of generality that X is the linearly dependent set. This means that there exist scalars a1, a2, ..., an, not all zero, and vectors x1, x2, ..., xn in X such that:

a1*x1 + a2*x2 + ... + an*xn = 0

Now, consider the set X ∪ Y. This set contains all vectors in X and Y. Therefore, the above non-trivial linear combination of vectors in X is also a non-trivial linear combination of vectors in X ∪ Y. Hence, X ∪ Y is linearly dependent.

(b) Consider the vector space R^3 over the field of real numbers. Let X = {(1, 0, 0)} and Y = {(0, 1, 0), (0, 0, 1)}. Both X and Y are linearly independent sets because no vector in either set can be written as a linear combination of the others. However, X ∪ Y = {(1, 0, 0), (0, 1, 0), (0, 0, 1)} is linearly dependent because the vector (1, 0, 0) can be written as a linear combination of the vectors (0, 1, 0) and (0, 0, 1).

3. (a) If X and Y are subsets of a vector space, and either X or Y is linearly dependent, show thatX ∪ Y is linearly dependent. [6 marks](b) Give an example of a vector space V with subsets X and Y , such that X and Y are linearlyindependent, but X ∪ Y is linearly dependent.

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