Question

if each vector ui or ei is in rn, which of the following is/are true? (select all that apply) group of answer choices if and , then {u1, u2, u3} is an orthogonal set. if is an orthogonal set, then s is linearly independent. if is linearly independent, then s is an orthogonal set. the set of standard vectors e1, e2, ..., en forms an orthogonal basis for rn. if s is an orthogonal set of n nonzero vectors in rn, then s is a basis for rn,

Answer 1

If each vector $u_i$ or $e_i$ is in $R^n$, the following statements are true: 1) If $u_1, u_2, u_3$ are orthogonal vectors, then they form an orthogonal set. 2) If a set $S$ is an orthogonal set, then it is linearly independent. 3) The set of standard vectors $e_1, e_2, ..., e_n$ forms an orthogonal basis for $R^n$.

Step-by-step explanation:

If each vector $u_i$ or $e_i$ is in $R^n$, the following statements are true:

1. If $u_1, u_2, u_3$ are orthogonal vectors, then they form an orthogonal set. An orthogonal set of vectors is a set of vectors that are all perpendicular to each other.
2. If a set $S$ is an orthogonal set, then it is linearly independent. This means that none of the vectors in $S$ can be written as a linear combination of the other vectors in $S$.
3. If a set $S$ is linearly independent, then it is an orthogonal set. This is not necessarily true. A set of vectors can be linearly independent without being orthogonal.
4. The set of standard vectors $e_1, e_2, ..., e_n$ forms an orthogonal basis for $R^n$. An orthogonal basis is a set of vectors that is both orthogonal and spans the entire vector space.
5. If $S$ is an orthogonal set of $n$ non-zero vectors in $R^n$, then $S$ is a basis for $R^n$. A basis for a vector space is a set of vectors that spans the entire space and is linearly independent.