Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 2: Problem 5
In Exercises 5-12, assume that \(T\) is a linear transformation. Refer to Example 7 for Exercises \(9-12\), if necessary. If \(T([1,0])=[3,-1]\) and \(T([0,1])=\) \([-2,5]\), find \(T([4,-6])\).
Short Answer
Expert verified
The transformation of \([4,-6]\) is \([24,-34]\).
Step by step solution
01
Understand the Definition of Linear Transformation
Recall that a linear transformation \( T : \mathbb{R}^n \to \mathbb{R}^m \) satisfies two key properties: additivity \( T(u + v) = T(u) + T(v) \) and homogeneity \( T(cu) = cT(u) \) for any vectors \( u, v \) and scalar \( c \). These properties allow us to find \( T([4, -6]) \) given \( T([1, 0]) \) and \( T([0, 1]) \).
02
Express \([4, -6]\) as a Linear Combination
The vector \([4, -6]\) can be expressed as a linear combination of the standard basis vectors. Specifically, \([4, -6] = 4[1,0] - 6[0,1]\).
03
Apply the Transformation Using the Linear Combination
Using the properties of linear transformations, compute \( T([4, -6]) = T(4[1,0] - 6[0,1]) = 4T([1,0]) - 6T([0,1]) \).
04
Calculate \( T([4, -6]) \) Using Given Values
Substitute the given vector transformations: \( T([1,0]) = [3,-1] \) and \( T([0,1]) = [-2,5] \). Calculate: \[ T([4,-6]) = 4[3,-1] - 6[-2,5] \] First, calculate each term: - \( 4[3,-1] = [12, -4] \) - \( 6[-2,5] = [-12, 30] \)Substitute these results: \[ T([4,-6]) = [12, -4] - [-12, 30] \]
05
Perform the Vector Subtraction
Compute the final transformation vector by subtracting the components: \[ T([4,-6]) = (12 - (-12), -4 - 30) = (12 + 12, -4 - 30) = [24, -34] \].
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.
Vector Space
A vector space is a collection of objects called vectors, which can be added together and multiplied by scalars to produce another vector. The concept is fundamental in linear algebra and provides a framework to carry out operations across many mathematical structures. In vector spaces:
- Vectors are elements within the space that can represent directions and magnitudes, or quantities with multiple elements, like in \( \mathbb{R}^n \).
- Scalar multiplication inherently scales a vector, changing its magnitude but not its direction.
- Vector addition combines vectors, producing another vector that encapsulates the direction influenced by both vectors.
Basis Vectors
Basis vectors are an essential tool in the architecture of vector spaces. Essentially, a basis is a set of vectors in a vector space that are linearly independent and span the vector space:
- Linearly independent means no vector in the set can be expressed as a combination of the other vectors in the set.
- Span implies that any vector in the space can be constructed as a linear combination of these basis vectors.
- For instance, in \( \mathbb{R}^2 \), the standard basis vectors are usually chosen as \([1, 0]\) and \([0, 1]\).
Additivity Property
The additivity property of linear transformations is an indication that these transformations preserve vector addition. This property states that for any vectors \( u \) and \( v \) in the vector space, the transformation \( T \) satisfies the condition \( T(u + v) = T(u) + T(v) \). This means:
- The transformation of a sum of vectors equals the sum of the transformations.
- This property ensures that lines and planes in space before transformation will remain linear after transformation.
- It's a critical aspect because it signifies that the operation of combining vectors through addition carries through under a linear transformation.
Homogeneity Property
The homogeneity property, also known as scalar compatibility, depicts how linear transformations interact with scalar multiplication. For a scalar \( c \) and a vector \( u \), the property is characterized by the equation \( T(cu) = cT(u) \). This implies:
- If you scale a vector before applying the transformation, it's the same as transforming the vector first and then applying the same scale afterward.
- This property maintains the notion of proportion among vectors and allows for broad manipulation while maintaining consistent directionality.
- In essence, if you imagine vectors as arrows pointing in space, the transformation doesn't affect their proportionality 鈥� merely their position.
