Question

Determine the dimension of each of the following vector spaces. (a) |[1,-2], [-2,4]| (b) |[-1,0,1], [1,2,3], [0,-1,0]| (c) |[1,2,3,4], [-1,1,0,2], [0,1,1,1]|

Answer 1

The dimensions of the given vector spaces are determined by finding a basis through row reduction. The dimensions for the vector spaces are 1 for (a), 3 for (b), and 3 for (c).

Step-by-step explanation:

To determine the dimension of each of the following vector spaces, we need to assess the linear independence of the given sets of vectors and find a basis for each space. A basis of a vector space is a set of vectors that are linearly independent and span the space. The dimension of the vector space is the number of vectors in the basis. For the vectors provided, we can perform the following steps:

1. Write the vectors as rows of a matrix.
2. Use Gaussian elimination to row-reduce the matrix to echelon form.
3. The non-zero rows after row reduction represent the linearly independent vectors.
4. Count the number of non-zero rows to obtain the dimension of the vector space.

Let's calculate the dimension for each vector space:

• (a) The set \(\\\) obviously contains multiples of each other, so there is only one linearly independent vector, thus the dimension is 1.
• (b) The set \(\\) requires row reduction to determine linear independence. Applying row reduction shows that the three vectors are linearly independent, therefore the dimension is 3.
• (c) The set \(\\) can be shown to be linearly independent through row reduction, resulting in a dimension of 3.

In conclusion, the dimensions of the vector spaces for each set are (a) 1, (b) 3, and (c) 3.