Question

For each of the situations described below,give an example(if it’s possible) or explain why it’s not possible.

a. A set of vectors that does not span R^3. After adding one more vector, the setdoes spanR3.
b. A set of vectors that are linearly dependent. After adding one more vector, theset becomes linearly independent.

Answer 1

A set of two vectors in R^3 can become a spanning set by adding a third independent vector. A linearly dependent set cannot become independent by adding more vectors. Vector addition is sensitive to magnitude and direction, but not to the order in which vectors are added.

Step-by-step explanation:

Vector Properties in Mathematics

For situation (a), an example of a set of vectors that does not span R^3 could be two vectors, such as (1,0,0) and (0,1,0). Since these vectors only cover the x- and y-axes, a third dimension is missing, and they do not fill the entire space of R^3. By adding another vector that is not in the span of the previous two, such as (0,0,1), the set now spans R^3.

For situation (b), it is not possible to have a set of vectors that are linearly dependent and become linearly independent by adding another vector. A new vector added to a linearly dependent set cannot remove existing dependencies; it can only potentially introduce more dependencies.

No, vectors with different magnitudes can never add to zero. Two vectors can add up to zero when they have the same magnitude but opposite directions. For three or more vectors, they can add up to zero if they form a closed polygon when placed tip-to-tail in sequence.

It is not possible to add a scalar to a vector because they are different mathematical objects; a scalar only has magnitude, while a vector has both magnitude and direction.

The order of addition of three vectors does not affect their sum due to the commutative and associative properties of vector addition. For any three vectors A, B, and C, the sum A + B + C will be the same as C + B + A or any other permutation of these vectors.

Answer 2

Step-by-step explanation:

a) If the given set of vectors does not span
`\mathbb{R}^(3)` , it means the number of linearly independent vectors are less than 3. So by adding one or more linearly independent vectors with respect to existing vectors, we can convert the set to basis and basis always spans vector space.

eg.
`S= \left \{ (1,0,0)^(T),(1,2,0)^(T) \right \}` this set does not span
`\mathbb{R}^(3)` . Since it has only two vectors and both vectors are linearly independent, so adding one linearly independent vector with respect to the vectors of S viz. (0,0,1)^{T} can span whole
`\mathbb{R}^(3)`.

b) It is not possible. Since if a set of vectors is linearly dependent then after adding linearly independent vectors it will also linearly dependent i.e. addition of a linearly independent vector does not effect on the set.