📚 Topic Summary
The standard basis for a vector space is a set of vectors that are linearly independent and span the entire space. This means that any vector in the space can be written as a unique linear combination of the basis vectors. For example, in $R^n$, the standard basis consists of vectors with a single '1' in one component and '0's elsewhere. Similarly, $P_n$ (polynomials of degree $n$ or less) has a standard basis of monomials, and $M_{m,n}$ (matrices of size $m \times n$) has a standard basis of matrices with a single '1' in one entry and '0's elsewhere. Understanding these standard bases is fundamental for performing operations and transformations in linear algebra.
This worksheet will help you practice identifying standard bases and understanding their properties. Let's dive in!
🧠 Part A: Vocabulary
Match each term with its definition. Write the corresponding number in the blank.
| Terms |
Definitions |
| 1. Vector Space |
a. A set of polynomials with degree less than or equal to $n$. |
| 2. Standard Basis |
b. A set of $m \times n$ matrices where each matrix has a single '1' and the rest are '0's. |
| 3. $R^n$ |
c. A set of vectors that spans the vector space and is linearly independent. |
| 4. $P_n$ |
d. A set that is closed under addition and scalar multiplication. |
| 5. $M_{m,n}$ |
e. A set of $n$-tuples. |
Answers: 1. __ 2. __ 3. __ 4. __ 5. __
✍️ Part B: Fill in the Blanks
The standard basis for $R^3$ is given by { (1, 0, 0), (0, 1, 0), (0, 0, 1) }. This set of vectors is both _______ _______ and spans $R^3$. Therefore, any vector in $R^3$ can be written as a _______ _______ of these basis vectors. The standard basis for $P_2$ is {1, x, $x^2$}. The dimension of $M_{2,3}$ is _______. The standard basis for $M_{2,3}$ will have _______ elements.
🤔 Part C: Critical Thinking
Explain why the standard basis is useful in linear algebra. Give at least two reasons. Provide examples to support your claims.
