Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 4: Problem 27
In Exercises \(27-30\) , use coordinate vectors to test the linear independence of the sets of polynomials. Explain your work. $$ 1+2 t^{3}, 2+t-3 t^{2},-t+2 t^{2}-t^{3} $$
Short Answer
Expert verified
The set of polynomials is linearly dependent.
Step by step solution
01
Define the Polynomials as Vectors
We need to express the polynomials as vectors. Each term in the polynomial corresponds to the coefficients of terms in a vector of dimension equal to the highest degree of the polynomial. Here the degree is 3.The polynomials are represented as vectors:1.\(1 + 0t + 0t^2 + 2t^3\) becomes \([1, 0, 0, 2]\).2. \(2 + 1t + (-3)t^2 + 0t^3\) becomes \([2, 1, -3, 0]\).3. \(0 + (-1)t + 2t^2 + (-1)t^3\) becomes \([0, -1, 2, -1]\).
02
Arrange Vectors into a Matrix
Place the vectors as columns of a matrix. This yields:\[A = \begin{bmatrix}1 & 2 & 0 \ 0 & 1 & -1 \ 0 & -3 & 2 \ 2 & 0 & -1 \end{bmatrix}\]
03
Row Reduce the Matrix
Apply row reduction (Gaussian elimination) to transform matrix \(A\) into row-echelon form. This simplifies finding dependencies between columns.After performing row reduction on the matrix \(A\), the row-echelon form might look like:\[\begin{bmatrix}1 & 0 & -1 \ 0 & 1 & -1 \ 0 & 0 & 0 \ 0 & 0 & 0 \end{bmatrix}\]
04
Analyze Linear Independence
Check the row-echelon form. If there's a pivot (leading 1) in each column, then the columns (vectors) are linearly independent.
In this matrix, there are leading 1's in the first two columns, but the third does not contain a pivot, indicating a dependency.
05
Conclude Dependency
Since there is no pivot in the third column in the row-echelon form, it indicates the set of vectors (polynomials) is linearly dependent.
This means some polynomial can be written as a linear combination of others.
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Coordinate Vectors
Coordinate vectors help us understand polynomials as sets of numbers, such as different "points" in space. Each coordinate in the vector represents the coefficient of a corresponding power of the variable (usually \(t\) in polynomials).
For instance, the polynomial \(1 + 2t^3\) can be expressed as the vector \([1, 0, 0, 2]\). Here, each position of the vector corresponds to the coefficients of \(t^0, t^1, t^2, t^3\) respectively.
Using coordinate vectors, we can easily apply linear algebra techniques to understand polynomial behavior.
For instance, the polynomial \(1 + 2t^3\) can be expressed as the vector \([1, 0, 0, 2]\). Here, each position of the vector corresponds to the coefficients of \(t^0, t^1, t^2, t^3\) respectively.
Using coordinate vectors, we can easily apply linear algebra techniques to understand polynomial behavior.
- Every vector is an ordered list of numbers, each representing a coefficient in the polynomial.
- This transformation allows us to use matrix operations to study polynomials.
Polynomial Representation as Vectors
Polynomials can be represented as vectors to simplify operations like addition, subtraction, and checking for dependencies.
The challenge lies in capturing each polynomial's coefficients in a form that softens algebraic manipulation. Consider the polynomial \(2 + t - 3t^2\), which is expressed as \([2, 1, -3, 0]\). Here, each number in the vector captures the coefficient of \(t^0\), \(t^1\), and so on.
Organizing polynomials this way helps in applying matrix operations. This form helps standardize methods for tackling complex algebraic expressions.
The challenge lies in capturing each polynomial's coefficients in a form that softens algebraic manipulation. Consider the polynomial \(2 + t - 3t^2\), which is expressed as \([2, 1, -3, 0]\). Here, each number in the vector captures the coefficient of \(t^0\), \(t^1\), and so on.
Organizing polynomials this way helps in applying matrix operations. This form helps standardize methods for tackling complex algebraic expressions.
- Each polynomial becomes a vector of numbers, rooted in its coefficients.
- This avails ways to analyze polynomials using vector operations rather than traditional algebraic methods.
Gaussian Elimination
Gaussian elimination is a technique used to simplify matrices, making them easier to interpret. When dealing with vectors in matrix form, we can apply Gaussian elimination to uncover any dependencies among them.
This involves performing operations to transform a matrix into its row-echelon form, where each row has more leading zeros than the row before.
Once in this form, it's simple to identify the pivot elements (the leading 1s) in each column.
This involves performing operations to transform a matrix into its row-echelon form, where each row has more leading zeros than the row before.
Once in this form, it's simple to identify the pivot elements (the leading 1s) in each column.
- We systematically use row operations to simplify matrices.
- Gaussian elimination helps reveal the structure and dependencies hidden within polynomial vectors.
Linear Dependence and Independence
Linear dependence and independence are important concepts when analyzing vectors. When vectors are linearly independent, no vector can be expressed as a combination of the others. Conversely, if there is a dependency, at least one vector can be expressed as such a combination.
During row reduction of matrix \(A\), linear dependence becomes obvious if not all columns have a pivot.
For instance, if a matrix of vectors results in a row-echelon form where some columns don't contain pivots, those vectors/forming polynomials are dependent.
During row reduction of matrix \(A\), linear dependence becomes obvious if not all columns have a pivot.
For instance, if a matrix of vectors results in a row-echelon form where some columns don't contain pivots, those vectors/forming polynomials are dependent.
- Linear independence signifies unique information and structure within vectors.
- Linear dependence signifies redundancy, where information encodes overlaps.
